The Cartan–Brauer Triangle in Modular Representation Theory

Chapter 2
The Cartan–Brauer Triangle

Let G be a finite group. Over any commutative ring R we have the group ring

R[G] = ag g : ag ∈ R
g∈G

with addition

ag g + bg g = (ag + bg )g
g∈G g∈G g∈G

and multiplication

ag g bg g = ah bh−1 g g.
g∈G g∈G g∈G h∈G

We fix an algebraically closed field k of characteristic p > 0. The representation
theory of G over k is the module theory of the group ring k[G]. This is our primary
object of study in the following.

2.1 The Setting

The main technical tool of our investigation will be a (0, p)-ring R for k which is a
complete local commutative integral domain R such that
– the maximal ideal mR ⊆ R is principal,
– R/mR = k, and
– the field of fractions of R has characteristic zero.

j
Exercise The only ideals of R are mR for j ≥ 0 and {0}.

P. Schneider, Modular Representation Theory of Finite Groups, 43
DOI 10.1007/978-1-4471-4832-6_2, © Springer-Verlag London 2013

44 2 The Cartan–Brauer Triangle

We note that there must exist an integer e ≥ 1—the ramification index of R—
such that Rp = me . There is, in fact, a canonical (0, p)-ring W (k) for k—its ring of
R
Witt vectors—with the additional property that mW (k) = W (k)p. Let K/K0 be any
finite extension of the field of fractions K0 of W (k). Then

R := a ∈ K : NormK/K0 (a) ∈ W (k)

is a (0, p)-ring for k with ramification index equal to [K : K0 ]. Proofs for all of this
can be found in §3–6 of [9].
In the following we fix a (0, p)-ring R for k. We denote by K the field of fractions
of R and by πR a choice of generator of mR , i.e. mR = RπR . The following three
group rings are now at our disposal:

K[G]

⊆
pr
R[G] k[G]

such that
K[G] = K ⊗R R[G] and k[G] = R[G]/πR R[G].
As explained before Proposition 1.7.4 there are the corresponding homomorphisms
between Grothendieck groups

K0 (K[G])

[P ]−→[K⊗R P ] κ
ρ
K0 (R[G]) K0 (k[G]).
[P ]−→[P /πR P ]

For the vertical arrow observe that, quite generally for any R[G]-module M, we
have
K[G] ⊗R[G] M = K ⊗R R[G] ⊗R[G] M = K ⊗R M.
We put
RK (G) := R K[G] and Rk (G) := R k[G] .
Since K has characteristic zero the group ring K[G] is semisimple, and we have

RK (G) = K0 K[G]

by Remark 1.7.3.iii. On the other hand, as a finite-dimensional k-vector space the
group ring k[G] of course is left and right artinian. In particular we have the Cartan

2.1 The Setting 45

homomorphism

cG : K0 k[G] −→ Rk (G)
[P ] −→ [P ].

Hence, so far, there is the diagram of homomorphisms

RK (G) Rk (G)

κ cG
ρ
K0 (R[G]) K0 (k[G]).

Clearly, R[G] is an R-algebra which is finitely generated as an R-module. Let us
collect some of what we know in this situation.
– (Proposition 1.3.6) R[G] is left and right noetherian, and any finitely generated
R[G]-module is complete as well as R[G]πR -adically complete.
– (Theorem 1.4.7) The Krull–Remak–Schmidt theorem holds for any finitely gen-
erated R[G]-module.
– (Proposition 1.5.5) 1 ∈ R[G] can be written as a sum of pairwise orthogonal
primitive idempotents; the set of all central idempotents in R[G] is finite; 1 is
equal to the sum of all primitive idempotents in Z(R[G]); any R[G]-module has
a block decomposition.
– (Proposition 1.5.7) For any idempotent ε ∈ k[G] there is an idempotent e ∈ R[G]
such that ε = e + R[G]πR .
– (Proposition 1.5.12) The projection map R[G] −→ k[G] restricts to a bijection
between the set of all central idempotents in R[G] and the set of all central idem-
potents in k[G]; in particular, the block decomposition of an R[G]-module M re-
duces modulo R[G]πR to the block decomposition of the k[G]-module M/πR M.
– (Proposition 1.6.10) Any finitely generated R[G]-module M has a projective
∼
=
cover P −→ M such that P / Jac(R[G])P − M/ Jac(R[G])M is an isomor-
→
phism.
We emphasize that R[G]πR ⊆ Jac(R[G]) by Lemma 1.3.5.iii. Moreover, the
ideal Jac(k[G]) = Jac(R[G])/R[G]πR in the left artinian ring k[G] is nilpotent by
Proposition 1.2.1.vi. We apply Proposition 1.7.4 to the rings R[G] and k[G] and we
see that the maps {P } −→ {P /πR P } −→ {P / Jac(R[G])P } induce the commuta-
tive diagram of bijections between finite sets

R[G] k[G]

R[G] = k[G]


as well as the commutative diagram of isomorphisms

∼
=
Z[R[G]] Z[k[G]]

∼
= ∼
=
∼
=
K0 (R[G]) K0 (k[G]).
ρ

For purposes of reference we state the last fact as a proposition.
∼
=
Proposition 2.1.1 The map ρ : K0 (R[G]) − K0 (k[G]) is an isomorphism; its in-
→
verse is given by sending [M] to the class of a projective cover of M as an R[G]-
module.

We define the composed map
ρ −1 κ
eG : K0 k[G] − → K0 R[G] − K0 K[G] = RK (G).
− →

Remark 2.1.2 Any finitely generated projective R-module is free.

Proof Since R is an integral domain 1 is the only idempotent in R. Hence the free
R-module R is indecomposable by Corollary 1.5.2. On the other hand, according to
Proposition 1.7.4.i the map
˜ ˆ
R −→ k
{P } −→ {P /πR P }
is bijective. Obviously, k is up to isomorphism the only simple k-module. Hence R is
up to isomorphism the only finitely generated indecomposable projective R-module.
An arbitrary finitely generated projective R-module P , by Lemma 1.1.6, is a fi-
nite direct sum of indecomposable ones. It follows that P must be isomorphic to
some R n .

2.2 The Triangle
We already have the two sides

RK (G) Rk (G)

eG cG

K0 (k[G])

of the triangle. To construct the third side we first introduce the following notion.

2.2 The Triangle 47

Definition Let V be a finite-dimensional K-vector space; a lattice L in V is an
R-submodule L ⊆ V for which there exists a K-basis e1 , . . . , ed of V such that

L = Re1 + · · · + Red .

Obviously, any lattice is free as an R-module. Furthermore, with L also aL, for
any a ∈ K × , is a lattice in V .

Lemma 2.2.1
i. Let L be an R-submodule of a K-vector space V ; if L is finitely generated then
L is free.
ii. Let L ⊆ V be an R-submodule of a finite-dimensional K-vector space V ; if L
is finitely generated as an R-module and L generates V as a K-vector space
then L is a lattice in V .
iii. For any two lattices L and L in V there is an integer m ≥ 0 such that
πR L ⊆ L .
m

Proof i. and ii. Let d ≥ 0 be the smallest integer such that the R-module L has d
generators e1 , . . . , ed . The R-module homomorphism

R d −→ L
(a1 , . . . , ad ) −→ a1 e1 + · · · + ad ed

is surjective. Suppose that (a1 , . . . , ad ) = 0 is an element in its kernel. Since at least
one ai is nonzero the integer
j
:= max j ≥ 0 : a1 , . . . , ad ∈ mR

is defined. Then ai = πR bi with bi ∈ R, and bi0 ∈ R × for at least one index
1 ≤ i0 ≤ d. Computing in the vector space V we have

0 = a1 e1 + · · · + ad ed = πR (b1 e1 + · · · + bd ed )

and hence
b1 e1 + · · · + bd ed = 0.
−1
But the latter equation implies ei0 = − i=i0 bi0 bi ei ∈ i=i0 Rei , which is a con-
tradiction to the minimality of d. It follows that the above map is an isomorphism.
This proves i. and, in particular, that

L = Re1 + · · · + Red .

For ii. it therefore suffices to show that e1 , . . . , ed , under the additional assumption
that L generates V , is a K-basis of V . This assumption immediately guarantees


that the e1 , . . . , ed generate the K-vector space V . To show that they are K-linearly
independent let

c1 e1 + · · · + cd ed = 0 with c1 , . . . , cd ∈ K.
j
We find a sufficiently large j ≥ 0 such that ai := πR ci ∈ R for any 1 ≤ i ≤ d. Then

0 = πR · 0 = πR (c1 e1 + · · · + cd ed ) = a1 e1 + · · · + ad ed .

By what we have shown above we must have ai = 0 and hence ci = 0 for any
1 ≤ i ≤ d.
iii. Let e1 , . . . , ed and f1 , . . . , fd be K-bases of V such that

L = Re1 + · · · + Red and L = Rf1 + · · · + Rfd .

We write
ej = c1j f1 + · · · + cdj fd with cij ∈ K,
and we choose an integer m ≥ 0 such that

πR cij ∈ R
m
for any 1 ≤ i, j ≤ d.

It follows that πR ej ∈ Rf1 + · · · + Rfd = L for any 1 ≤ j ≤ d and hence that
m

πRmL ⊆ L .

Suppose that V is a finitely generated K[G]-module. Then V is finite-dimension-
al as a K-vector space. A lattice L in V is called G-invariant if we have gL ⊆ L
for any g ∈ G. In particular, L is a finitely generated R[G]-submodule of V , and
L/πR L is a k[G]-module of finite length.

Lemma 2.2.2 Any finitely generated K[G]-module V contains a lattice which is
G-invariant.

Proof We choose a basis e1 , . . . , ed of the K-vector space V . Then L := Re1 +
· · · + Red is a lattice in V . We define the R[G]-submodule

L := gL
g∈G

of V . With L also L generates V as a K-vector space. Moreover, L is finitely
generated by the set {gei : 1 ≤ i ≤ d, g ∈ G} as an R-module. Therefore, by
Lemma 2.2.1.ii, L is a G-invariant lattice in V .

Whereas the K[G]-module V always is projective by Remark 1.7.3 a G-invariant
lattice L in V need not to be projective as an R[G]-module. We will encounter an
example of this later on.

2.2 The Triangle 49

Theorem 2.2.3 Let L and L be two G-invariant lattices in the finitely generated
K[G]-module V ; we then have

[L/πR L] = L /πR L in Rk (G).

Proof We begin by observing that, for any a ∈ K × , the map
∼
=
L/πR L −→ (aL)/πR (aL)
x + πR L −→ ax + πR (aL)

is an isomorphism of k[G]-modules, and hence

[L/πR L] = (aL)/πR (aL) in Rk (G).
m
By applying Lemma 2.2.1.iii (and replacing L by πR L for some sufficiently large
m ≥ 0) we therefore may assume that L ⊆ L . By applying Lemma 2.2.1.iii again
to L and L (while interchanging their roles) we find an integer n ≥ 0 such that

πR L ⊆ L ⊆ L .
n

We now proceed by induction with respect to n. If n = 1 we have the two exact
sequences of k[G]-modules

0 −→ L/πR L −→ L /πR L −→ L /L −→ 0

and
0 −→ πR L /πR L −→ L/πR L −→ L/πR L −→ 0.
It follows that

L /πR L = L/πR L + L /L = L/πR L + πR L /πR L
= L/πR L + [L/πR L] − L/πR L
= [L/πR L]

in Rk (G). For n ≥ 2 we consider the R[G]-submodule

M := πR L + L.
n−1

It is a G-invariant lattice in V by Lemma 2.2.1.ii and satisfies

πR L ⊆ M ⊆ L
n−1
and πR M ⊆ L ⊆ M.

Applying the case n = 1 to L and M we obtain [M/πR M] = [L/πR L]. The induc-
tion hypothesis for M and L gives [L /πR L ] = [M/πR M].


The above lemma and theorem imply that

Z[MK[G] ] −→ Rk (G)
{V } −→ [L/πR L],

where L is any G-invariant lattice in V , is a well-defined homomorphism. If V1
and V2 are two finitely generated K[G]-modules and L1 ⊆ V1 and L2 ⊆ V2 are
G-invariant lattices then L1 ⊕ L2 is a G-invariant lattice in V1 ⊕ V2 and

(L1 ⊕ L2 )/πR (L1 ⊕ L2 ) = [L1 /πR L1 ⊕ L2 /πR L2 ]
= [L1 /πR L1 ] + [L2 /πR L2 ].

It follows that the subgroup Rel ⊆ Z[MK[G] ] lies in the kernel of the above map so
that we obtain the homomorphism

dG : RK (G) −→ Rk (G)
[V ] −→ [L/πR L].

It is called the decomposition homomorphism of G. The Cartan–Brauer triangle is
the diagram

dG
RK (G) Rk (G)

eG cG

K0 (k[G]).

Lemma 2.2.4 The Cartan–Brauer triangle is commutative.

Proof Let P be a finitely generated projective R[G]-module. We have to show that

dG κ [P ] = cG ρ [P ]

holds true. By definition the right-hand side is equal to [P /πR P ] ∈ Rk (G). More-
over, κ([P ]) = [K ⊗R P ] ∈ RK (G). According to Proposition 1.6.4 the R(G)-
module P is a direct summand of a free R[G]-module. But R[G] and hence any free
R[G]-module also is free as an R-module. We see that P as an R-module is finitely
generated projective and hence free by Remark 2.1.2. We conclude that P ∼ R d is a
=
G-invariant lattice in the K[G]-module K ⊗R P ∼ K ⊗R R d = (K ⊗R R)d = K d ,
=
and we obtain dG (κ([P ])) = dG ([K ⊗R P ]) = [P /πR P ].

Let us consider two “extreme” situations where the maps in the Cartan–Brauer
triangle can be determined completely. First we look at the case where p does not
divide the order |G| of the group G. Then k[G] is semisimple. Hence we have
k[G] = k[G] and K0 (k[G]) = Rk (G) by Remark 1.7.3. The map cG , in particular,
is the identity.

2.2 The Triangle 51

Proposition 2.2.5 If p |G| then any R[G]-module M which is projective as an
R-module also is projective as an R[G]-module.

Proof We consider any “test diagram” of R[G]-modules

M

α
β
L N 0.

Viewing this as a “test diagram” of R-modules our second assumption ensures the
existence of an R-module homomorphism α0 : M −→ L such that β ◦ α0 = α. Since
|G| is a unit in R by our first assumption, we may define a new R-module homo-
morphism α : M −→ L by
˜

α(x) := |G|−1
˜ gα0 g −1 x for any x ∈ M.
g∈G

˜
One easily checks that α satisfies

α(hx) = hα(x)
˜ ˜ for any h ∈ G and any x ∈ M.

˜
This means that α is, in fact, an R[G]-module homomorphism. Moreover, we com-
pute

β α(x) = |G|−1
˜ gβ α0 g −1 x = |G|−1 gα g −1 x
g∈G g∈G

= |G|−1 gg −1 α(x) = α(x).
g∈G

Corollary 2.2.6 If p |G| then all three maps in the Cartan–Brauer triangle are
isomorphisms; more precisely, we have the triangle of bijections

K[G] k[G]

{P }−→{K⊗R P } {P }−→{P /πR P }

R[G].

Proof We already have remarked that cG is the identity. Hence it suffices to show
that the map κ : K0 (R[G]) −→ RK (G) is surjective. Let V be any finitely generated
K[G]-module. By Lemma 2.2.2 we find a G-invariant lattice L in V . It satisfies
V = K ⊗R L by definition. Proposition 2.2.5 implies that L is a finitely generated


projective R[G]-module. We conclude that [L] ∈ K0 (R[G]) with κ([L]) = [V ].
This argument in fact shows that the map

MR[G] / ∼ −→ MK[G] / ∼
= =
{P } −→ {K ⊗R P }

is surjective. Let P and Q be two ﬁnitely generated projective R[G]-modules such
that K ⊗R P ∼ K ⊗R Q as K[G]-modules. The commutativity of the Cartan–Brauer
=
triangle then implies that

[P /πR P ] = dG [K ⊗R P ] = dG [K ⊗R Q] = [Q/πR Q].

Let P = P1 ⊕ · · · ⊕ Ps and Q = Q1 ⊕ · · · ⊕ Qt be decompositions into indecom-
posable submodules. Then
s t
P /πR P = Pi /πR Pi and Q/πR Q = Qj /πR Qj
i=1 j =1

are decompositions into simple submodules. Using Proposition 1.7.1 the identity
s t
[Pi /πR Pi ] = [P /πR P ] = [Q/πR Q] = [Qj /πR Qj ]
i=1 j =1

implies that s = t and that there is a permutation σ of {1, . . . , s} such that

Qj /πR Qj ∼ Pσ (j ) /πR Pσ (j )
= for any 1 ≤ j ≤ s

as k[G]-modules. Applying Proposition 1.7.4.i we obtain

Qj ∼ Pσ (j )
= for any 1 ≤ j ≤ s

as R[G]-modules. It follows that P ∼ Q as R[G]-modules. Hence the above map
=
between sets of isomorphism classes is bijective. Obviously, if K ⊗R P is in-
decomposable (i.e. simple) then P was indecomposable. Vice versa, if P is in-
decomposable then the above reasoning says that P /πR P is simple. Because of
[P /πR P ] = dG ([K ⊗R P ]) it follows that K ⊗R P must be indecomposable.

The second case is where G is a p-group. For a general group G we have the
ring homomorphism

k[G] −→ k

ag g −→ ag
g∈G g∈G

2.2 The Triangle 53

which is called the augmentation of k[G]. It makes k into a simple k[G]-module
which is called the trivial k[G]-module. Its kernel is the augmentation ideal

Ik [G] := ag g ∈ k[G] : ag = 0 .
g∈G g∈G

Proposition 2.2.7 If G is a p-group then we have Jac(k[G]) = Ik [G]; in particular,
k[G] is a local ring and the trivial k[G]-module is, up to isomorphism, the only
simple k[G]-module.

Proof We will prove by induction with respect to the order |G| = p n of G that the
trivial module, up to isomorphism, is the only simple k[G]-module. There is nothing
to prove if n = 0. We therefore suppose that n ≥ 1. Note that we have
n n n
gp = 1 and hence (g − 1)p = g p − 1 = 1 − 1 = 0

for any g ∈ G. The center of a nontrivial p-group is nontrivial. Let g0 = 1 be a
central element in G. We now consider any simple k[G]-module M and we denote
by π : k[G] −→ Endk (M) the corresponding ring homomorphism. Then
pn n n
π(g0 ) − idM = π(g0 − 1)p = π (g0 − 1)p = π(0) = 0.

Since g0 is central (π(g0 ) − idM )(M) is a k[G]-submodule of M. But M is simple.
Hence (π(g0 ) − idM )(M) = {0} or = M. The latter would inductively imply that
n
(π(g0 ) − idM )p (M) = M = {0} which is contradiction. We obtain π(g0 ) = idM
which means that the cyclic subgroup g0 is contained in the kernel of the group
homomorphism π : G −→ Autk (M). Hence we have a commutative diagram of
group homomorphisms

π
G Autk (M)

pr π
G/ g0 .

We conclude that M already is a simple k[G/ g0 ]-module and therefore is the
trivial module by the induction hypothesis.
The identity Jac(k[G]) = Ik [G] now follows from the deﬁnition of the Jacobson
radical, and Proposition 1.4.1 implies that k[G] is a local ring.

Suppose that G is a p-group. Then Propositions 2.2.7 and 1.7.1 together imply
that the map
∼
=
Z −→ Rk (G)
m −→ m[k]


is an isomorphism. To compute the inverse map let M be a k[G]-module of finite
length, and let {0} = M0 M1 · · · Mt = M be a composition series. We must
have Mi /Mi−1 ∼ k for any 1 ≤ i ≤ t . It follows that
=
t t
[M] = [Mi /Mi−1 ] = t · [k] and dimk M = dimk Mi /Mi−1 = t.
i=1 i=1

Hence the inverse map is given by
[M] −→ dimk M.
Furthermore, from Proposition 1.7.4 we obtain that
∼
=
Z −→ K0 R[G]
m −→ m R[G]
is an isomorphism. Because of dimk k[G] = |G| the Cartan homomorphism, under
these identifications, becomes the map
cG : Z −→ Z
m −→ m · |G|.
For any finitely generated K[G]-module V and any G-invariant lattice L in V we
have dimK V = dimk L/πR L. Hence the decomposition homomorphism becomes
dG : RK (G) −→ Z
[V ] −→ dimK V .
Altogether the Cartan–Brauer triangle of a p-group G is of the form

[V ]−→dimK V
RK (G) Z

1−→[K[G]] ·|G|
Z.

We also see that the trivial K[G]-module K has the G-invariant lattice R which
cannot be projective as an R[G]-module if G = {1}.
Before we can establish the finer properties of the Cartan–Brauer triangle we
need to develop the theory of induction.

2.3 The Ring Structure of RF (G), and Induction
In this section we let F be an arbitrary field, and we consider the group ring F [G]
and its Grothendieck group RF (G) := R(F [G]).

2.3 The Ring Structure of RF (G), and Induction 55

Let V and W be two (finitely generated) F [G]-modules. The group G acts on
the tensor product V ⊗F W by

g(v ⊗ w) := gv ⊗ gw for v ∈ V and w ∈ W.

In this way V ⊗F W becomes a (finitely generated) F [G]-module, and we obtain
the multiplication map

Z[MF [G] ] × Z[MF [G] ] −→ Z[MF [G] ]
{V }, {W } −→ {V ⊗F W }.

Since the tensor product, up to isomorphism, is associative and commutative this
multiplication makes Z[MF [G] ] into a commutative ring. Its unit element is the
isomorphism class {F } of the trivial F [G]-module.

Remark 2.3.1 The subgroup Rel is an ideal in the ring Z[MF [G] ].

Proof Let V be a (finitely generated) F [G]-module and let

α β
0 −→ L − M − N −→ 0
→ →

be a short exact sequence of F [G]-modules. We claim that the sequence

idV ⊗α idV ⊗β
0 −→ V ⊗F L − − → V ⊗F M − − → V ⊗F N −→ 0
−− −−

is exact as well. This shows that the subgroup Rel is preserved under multiplication
by {V }. The exactness in question is purely a problem about F -vector spaces. But as
vector spaces we have M ∼ L ⊕ N and hence V ⊗F M ∼ (V ⊗F L) ⊕ (V ⊗F N ).
= =

It follows that RF (G) naturally is a commutative ring with unit element [F ] such
that
[V ] · [W ] = [V ⊗F W ].
Let H ⊆ G be a subgroup. Then F [H ] ⊆ F [G] is a subring (with the same unit
element). Any F [G]-module V , by restriction of scalars, can be viewed as an F [H ]-
module. If V is finitely generated as an F [G]-module then V is a finite-dimensional
F -vector space and, in particular, is finitely generated as an F [H ]-module. Hence
we have the ring homomorphism

resG : RF (G) −→ RF (H )
H
[V ] −→ [V ].

On the other hand, for any F [H ]-module W we have, by base extension, the F [G]-
module F [G] ⊗F [H ] W . Obviously, the latter is finitely generated over F [G] if the


former was ﬁnitely generated over F [H ]. The ﬁrst Frobenius reciprocity says that
∼
=
HomF [G] F [G] ⊗F [H ] W, V −→ HomF [H ] (W, V )
α −→ w −→ α(1 ⊗ w)

is an F -linear isomorphism for any F [H ]-module W and any F [G]-module V .

α β
Remark 2.3.2 For any short exact sequence 0 −→ L − M − N −→ 0 of F [H ]-
→ →
modules the sequence of F [G]-modules
idF [G] ⊗α idF [G] ⊗β
0 → F [G] ⊗F [H ] L − − − → F [G] ⊗F [H ] M − − − → F [G] ⊗F [H ] N → 0
−−− −−−

is exact as well.

Proof Let g1 , . . . , gr ∈ G be a set of representatives for the left cosets of H in G.
Then g1 , . . . , gr also is a basis of the free right F [H ]-module F [G]. It follows that,
for any F [H ]-module W , the map
∼
=
W r −→ F [G] ⊗F [H ] W
(w1 , . . . , wr ) −→ g1 ⊗ w1 + · · · + gr ⊗ wr

is an F -linear isomorphism. We see that, as a sequence of F -vector spaces, the
sequence in question is just the r-fold direct sum of the original exact sequence
with itself.

Remark 2.3.3 For any F [H ]-module W , if the F [G]-module F [G] ⊗F [H ] W is
simple then W is a simple F [H ]-module.

Proof Let W ⊆ W be any F [H ]-submodule. Then F [G] ⊗F [H ] W is an F [G]-
submodule of F [G] ⊗F [H ] W by Remark 2.3.2. Since the latter is simple we must
have F [G] ⊗F [H ] W = {0} or = F [G] ⊗F [H ] W . Comparing dimensions using the
argument in the proof of Remark 2.3.2 we obtain

[G : H ] · dimF W = dimF F [G] ⊗F [H ] W = 0 or
= dimF F [G] ⊗F [H ] W = [G : H ] · dimF W.

We see that dimF W = 0 or = dimF W and therefore that W = {0} or = W . This
proves that W is a simple F [H ]-module.

It follows that the map

Z[MF [H ] ] −→ Z[MF [G] ]
{W } −→ F [G] ⊗F [H ] W

2.3 The Ring Structure of RF (G), and Induction 57

preserves the subgroups Rel in both sides and therefore induces an additive homo-
morphism
indG :
H RF (H ) −→ RF (G)
[W ] −→ F [G] ⊗F [H ] W

(which is not multiplicative!).

Proposition 2.3.4 We have
indG (y) · x = indG y · resG (x)
H H H for any x ∈ RF (G) and y ∈ RF (H ).

Proof It sufﬁces to show that, for any F [G]-module V and any F [H ]-module W ,
we have an isomorphism of F [G]-modules
F [G] ⊗F [H ] W ⊗F V ∼ F [G] ⊗F [H ] (W ⊗F V ).
=

One checks (exercise!) that such an isomorphism is given by
(g ⊗ w) ⊗ v −→ g ⊗ w ⊗ g −1 v .

Corollary 2.3.5 The image of indG : RF (H ) −→ RF (G) is an ideal in RF (G).
H

We also mention the obvious transitivity relations
resH ◦ resG = resG
H H H and indG ◦ indH = indG
H H H

for any chain of subgroups H ⊆ H ⊆ G.
An alternative way to look at induction is the following. Let W be any F [H ]-
module. Then
IndG (W ) := φ : G −→ W : φ(gh) = h−1 φ(g) for any g ∈ G, h ∈ H
H

equipped with the left translation action of G given by
g
φ g := φ g −1 g

is an F [G]-module called the module induced from W . But, in fact, the map
∼
=
F [G] ⊗F [H ] W −→ IndG (W )
H

ag g ⊗ w −→ φ g := ag h hw
g∈G h∈H

is an isomorphism of F [G]-modules. This leads to the second Frobenius reciprocity
isomorphism
∼
=
HomF [G] V , IndG (W ) −→ HomF [H ] (V , W )
H

α −→ v −→ α(v)(1) .


We also need to recall the character theory of G in the semisimple case. For this
we assume for the rest of this section that the order of G is prime to the characteristic
of the field F . Any finitely generated F [G]-module V is a finite-dimensional F -
vector space. Hence we may introduce the function

χV : G −→ F
g
g −→ tr(g; V ) = trace of V − V
→

which is called the character of V . It depends only on the isomorphism class of V .
Characters are class functions on G, i.e. they are constant on each conjugacy class
of G. For any two finitely generated F [G]-modules V1 and V2 we have

χV1 ⊕V2 = χV1 + χV2 and χV1 ⊗F V2 = χV1 · χV2 .

Let Cl(G, F ) denote the F -vector space of all class functions G −→ F . By point-
wise multiplication of functions it is a commutative F -algebra. The above identities
imply that the map

Tr : RF (G) −→ Cl(G, F )
[V ] −→ χV

is a ring homomorphism.

Definition The field F is called a splitting field for G if, for any simple F [G]-
module V , we have EndF [G] (V ) = F .

If F is algebraically closed then it is a splitting field for G.

Theorem 2.3.6
i. If the field F has characteristic zero then the characters {χV : {V } ∈ F [G]} are
F -linearly independent.
ii. If F is a splitting field for G then the characters {χV : {V } ∈ F [G]} form a basis
of the F -vector space Cl(G, F ).
iii. If F has characteristic zero then two finitely generated F [G]-modules V1 and
V2 are isomorphic if and only if χV1 = χV2 holds true.

Corollary 2.3.7
i. If F has characteristic zero then the map Tr is injective.
ii. If F is a splitting field for G then the map Tr induces an isomorphism of
F -algebras
∼
=
F ⊗Z RF (G) −→ Cl(G, F ).

2.4 The Burnside Ring 59

iii. If F has characteristic zero then the map

MF [G] / ∼ −→ Cl(G, F )
=
{V } −→ χV

is injective.

Proof For i. and ii. use Proposition 1.7.1.

2.4 The Burnside Ring
A G-set X is a set equipped with a G-action

G × X −→ X
(g, x) −→ gx

such that

1x = x and g(hx) = (gh)x for any g, h ∈ G and any x ∈ X.

Let X and Y be two G-sets. Their disjoint union X ∪ Y is a G-set in an obvious
way. But also their cartesian product X × Y is a G-set with respect to

g(x, y) := (gx, gy) for (x, y) ∈ X × Y.

We will call X and Y isomorphic if there is a bijective map α : X −→ Y such that
α(gx) = gα(x) for any g ∈ G and x ∈ X.
Let SG denote the set of all isomorphism classes {X} of finite G-sets X. In the
free abelian group Z[SG ] we consider the subgroup Rel generated by all elements
of the form

{X ∪ Y } − {X} − {Y } for any two finite G-sets X and Y .

We define the factor group

B(G) := Z[SG ]/ Rel,

and we let [X] ∈ B(G) denote the image of the isomorphism class {X}. In fact, the
map

Z[SG ] × Z[SG ] −→ Z[SG ]
{X}, {Y } −→ {X × Y }

makes Z[SG ] into a commutative ring in which the unit element is the isomorphism
class of the G-set with one point. Because of (X1 ∪ X2 ) × Y = (X1 × Y ) ∪ (X2 × Y )


the subgroup Rel is an ideal in Z[SG ]. We see that B(G) is a commutative ring. It
is called the Burnside ring of G.
Two elements x, y in a G-set X are called equivalent if there is a g ∈ G such
that y = gx. This defines an equivalence relation on X. The equivalence classes
are called G-orbits. They are of the form Gx = {gx : g ∈ G} for some x ∈ X.
A nonempty G-set which consists of a single G-orbit is called simple (or transi-
tive or a principal homogeneous space). The decomposition of an arbitrary G-set X
into its G-orbits is the unique decomposition of X into simple G-sets. In particular,
the only G-subsets of a simple G-set Y are Y and ∅. We let SG denote the set of
isomorphism classes of simple G-sets.
∼
=
Lemma 2.4.1 Z[SG ] −→ B(G).

Proof If X = Y1 ∪ · · · ∪ Yn is the decomposition of X into its G-orbits then we put
π {X} := {Y1 } + · · · + {Yn }.

This defines an endomorphism π of Z[SG ] which is idempotent with im(π) =
Z[SG ]. It is rather clear that Rel ⊆ ker(π). Moreover, using the identities
[Y1 ] + [Y2 ] = [Y1 ∪ Y2 ], [Y1 ∪ Y2 ] + [Y3 ] = [Y1 ∪ Y2 ∪ Y3 ], ...,
[Y1 ∪ · · · ∪ Yn−1 ] + [Yn ] = [X]

we see that
[X] = [Y1 ] + · · · + [Yn ]
and hence that {X} − π({X}) ∈ Rel. As in the proof of Proposition 1.7.1 these three
facts together imply
Z[SG ] = Z[SG ] ⊕ Rel .

For any subgroup H ⊆ G the coset space G/H is a simple G-set with respect to
G × G/H −→ G/H
g, g H −→ gg H.

Simple G-sets of this form are called standard G-sets.

Remark 2.4.2 Each simple G-set X is isomorphic to some standard G-set G/H .

Proof We fix a point x ∈ X. Let Gx := {g ∈ G : gx = x} be the stabilizer of x in G.
Then

G/Gx −→ X
gGx −→ gx

is an isomorphism.


It follows that the set SG is finite.

Lemma 2.4.3 Two standard G-sets G/H1 and G/H2 are isomorphic if and only if
−1
there is a g0 ∈ G such that g0 H1 g0 = H2 .

−1
Proof If g0 H1 g0 = H2 then

G/H1 −→ G/H2
gH1 −→ gg0 H2
is an isomorphism of G-sets. Vice versa, let

α: G/H1 −→ G/H2
be an isomorphism of G-sets. We have α(1H1 ) = g0 H2 for some g0 ∈ G and then
g0 H2 = α(1H1 ) = α(h1 H1 ) = h1 α(1H1 ) = h1 g0 H2
−1
for any h1 ∈ H1 . This implies g0 H1 g0 ⊆ H2 . On the other hand
−1 −1
g0 H1 = α −1 (1H2 ) = α −1 (h2 H2 ) = h2 α −1 (1H2 ) = h2 g0 H1
−1
for any h2 ∈ H2 which implies g0 H2 g0 ⊆ H1 .

Exercise 2.4.4 Let G/H1 and G/H2 be two standard G-sets; we then have
[G/H1 ] · [G/H2 ] = G/H1 ∩ gH2 g −1 in B(G)
g∈H1 G/H2

where H1 G/H2 denotes the space of double cosets H1 gH2 in G.

Let F again be an arbitrary field. For any finite set X we have the finite-
dimensional F -vector space

F [X] := ax x : ax ∈ F
x∈X

“with basis X”. Suppose that X is a finite G-set. Then the group G acts on F [X] by

g ax x := ax gx = ag −1 x x.
x∈X x∈X x∈X

In this way F [X] becomes a finitely generated F [G]-module (called a permutation
module). If α : X −→ Y is an isomorphism of finite G-sets then
∼
=
α: F [X] −→ F [Y ]
˜
ax x −→ ax α(x) = aα −1 (y) y
x∈X x∈X y∈Y


is an isomorphism of F [G]-modules. It follows that the map

SG −→ MF [G] / ∼
=
{X} −→ F [X]

is well defined. We obviously have

F [X1 ∪ X2 ] = F [X1 ] ⊕ F [X2 ]

for any two finite G-sets X1 and X2 . Hence the above map respects the subgroups
Rel in both sides and induces a group homomorphism

b: B(G) −→ RF (G)
[X] −→ F [X] .

Remark There is a third interesting Grothendieck group for the ring F [G] which is
the factor group
AF (G) := Z[MF [G] ]/ Rel⊕
with respect to the subgroup Rel⊕ generated by all elements of the form

{M ⊕ N } − {M} − {N }

where M and N are arbitrary finitely generated F [G]-modules. We note that Rel⊕ ⊆
Rel. The above map b is the composite of the maps
pr
B(G) −→ AF (G) − RF (G)
→
[X] −→ F [X] −→ F [X] .

Remark 2.4.5 For any two finite G-sets X1 and X2 we have

F [X1 × X2 ] ∼ F [X1 ] ⊗F F [X2 ]
=

as F [G]-modules.

Proof The vectors (x1 , x2 ), resp. x1 ⊗ x2 , for x1 ∈ X1 and x2 ∈ X2 , form an F -basis
of the left-, resp. right-, hand side. Hence there is a unique F -linear isomorphism
mapping (x1 , x2 ) to x1 ⊗ x2 . Because of

g(x1 , x2 ) = (gx1 , gx2 ) −→ gx1 ⊗ gx2 = g(x1 ⊗ x2 ),

for any g ∈ G, this map is an F [G]-module isomorphism.

It follows that the map

b: B(G) −→ RF (G)


is a ring homomorphism. Note that the unit element [G/G] in B(G) is mapped to
the class [F ] of the trivial module F which is the unit element in RF (G).

Lemma 2.4.6
i. For any standard G-set G/H we have

b [G/H ] = indG (1)
H

where 1 on the right-hand side denotes the unit element of RF (H ).
ii. For any finite G-set X we have

tr g; F [X] = {x ∈ X : gx = x} ∈ F for any g ∈ G.

Proof i. Let F be the trivial F [H ]-module. It suffices to establish an isomorphism

F [G/H ] ∼ F [G] ⊗F [H ] F.
=

For this purpose we consider the F -bilinear map

β: F [G] × F −→ F [G/H ]

ag g, a −→ aag gH.
g∈G g∈G

Because of
β(gh, a) = ghH = gH = β(g, a) = β(g, ha)
for any h ∈ H the map β is F [H ]-balanced and therefore induces a well-defined
F -linear map
˜
β: F [G] ⊗F [H ] F −→ F [G/H ]

ag g ⊗ a −→ aag gH.
g∈G g∈G

As discussed in the proof of Remark 2.3.2 we have F [G] ⊗F [H ] F ∼ F [G:H ] as F -
=
˜
vector spaces. Hence β is a map between F -vector spaces of the same dimension.
It obviously is surjective and therefore bijective. Finally the identity

˜
β g ag g ⊗ a ˜
=β ag g g ⊗ a = aag g gH
g∈G g∈G g∈G

=g aag gH ˜
=g β ag g ⊗ a
g∈G g∈G

˜
for any g ∈ G shows that β is an isomorphism of F [G]-modules.


g·
ii. The matrix (ax,y )x,y of the F -linear map F [X] − F [X] with respect to the
→
basis X is given by the equations

gy = ax,y x.
x∈X

But gy ∈ X and hence
1 if x = gy
ax,y =
0 otherwise.
It follows that

tr g; F [X] = ax,x = 1 = {x ∈ X : gx = x} ∈ F.
x∈X gx=x

Remark
1. The map b rarely is injective. Let G = S3 be the symmetric group on three letters.
It has four conjugacy classes of subgroups. Using Lemma 2.4.1, Remark 2.4.2,
and Lemma 2.4.3 it therefore follows that B(S3 ) ∼ Z4 . On the other hand, S3
=
has only three conjugacy classes of elements. Hence Proposition 1.7.1 and The-
orem 2.3.6.ii imply that RC (S3 ) ∼ Z3 .
=
2. In general the map b is not surjective either. But there are many structural results
about its cokernel. For example, the Artin induction theorem implies that

|G| · RQ (G) ⊆ im(b).

We therefore introduce the subring

PF (G) := im(b) ⊆ RF (G).

Let H be a family of subgroups of G with the property that if H ⊆ H is a subgroup
of some H ∈ H then also H ∈ H. We introduce the subgroup B(G, H) ⊆ B(G)
generated by all [G/H ] for H ∈ H as well as its image PF (G, H) ⊆ PF (G) under
the map b.

Lemma 2.4.7 B(G, H) is an ideal in B(G), and hence PF (G, H) is an ideal in
PF (G).

Proof We have to show that, for any H1 ∈ H and any subgroup H2 ⊆ G, the element
[G/H1 ] · [G/H2 ] lies in B(G, H). This is immediately clear from Exercise 2.4.4.
But a less detailed argument sufﬁces. Obviously [G/H1 ] · [G/H2 ] is the sum of the
−1
classes of the G-orbits in G/H1 × G/H2 . Let G(g1 H1 , g2 H2 ) = G(H1 , g1 g2 H2 )
−1
be such a G-orbit. The stabilizer H ⊆ G of the element (H1 , g1 g2 H2 ) is contained
in H1 and therefore belongs to H. It follows that

G(g1 H1 , g2 H2 ) = G/H ∈ B(G, H).


Definition
i. Let be a prime number. A finite group H is called -hyper-elementary if it
contains a cyclic normal subgroup C such that |C| and H /C is an -group.
ii. A finite group is called hyper-elementary if it is -hyper-elementary for some
prime number .

Exercise Let H be an -hyper-elementary group. Then:
i. Any subgroup of H is -hyper-elementary;
ii. let C ⊆ H be a cyclic normal subgroup as in the definition, and let L ⊆ H be any
-Sylow subgroup; then the map C × L −→ H sending (c, g) to cg is a bijection
of sets.

Let Hhe denote the family of hyper-elementary subgroups of G. By the exercise
Lemma 2.4.7 is applicable to Hhe .

Theorem 2.4.8 (Solomon) Suppose that F has characteristic zero; then

PF (G, Hhe ) = PF (G).

Proof Because of Lemma 2.4.7 it suffices to show that the unit element 1 ∈ PF (G)
already lies in PF (G, Hhe ). According to Lemma 2.4.6.ii the characters

χF [X] (g) = {x ∈ X : gx = x} ∈ Z,

for any g ∈ G and any finite G-set X, have integral values. Hence we have the
well-defined ring homomorphisms

tg : PF (G) −→ Z
z −→ Tr(z)(g)

for g ∈ G. On the one hand, by Corollary 2.3.7.i, they satisfy

ker(tg ) = ker Tr |PF (G) = {0}. (2.4.1)
g∈G

On the other hand, we claim that

tg PF (G, Hhe ) = Z (2.4.2)

holds true for any g ∈ G. We fix a g0 ∈ G in the following. Since the image
tg0 (PF (G, Hhe )) is an additive subgroup of Z and hence is of the form nZ for some
n ≥ 0 it suffices to find, for any prime number , an -hyper-elementary subgroup
H ⊆ G such that the integer


tg0 F [G/H ] = χF [G/H ] (g0 ) = {x ∈ G/H : g0 x = x}
= gH ∈ G/H : g −1 g0 g ∈ H

is not contained in Z. We also ﬁx .
The wanted -hyper-elementary subgroup H will be found in a chain of sub-
groups
C ⊆ g0 ⊆ H ⊆ N
with C being normal in N which is constructed as follows. Let n ≥ 1 be the order
of g0 , and write n = s m with l m. The cyclic subgroup g0 ⊆ G generated by g0
then is the direct product
s
g0 = g0 × g0
m

s
where g0 is an -group and C := g0 is a cyclic group of order prime to .
m

We deﬁne N := {g ∈ G : gCg −1 = C} to be the normalizer of C in G. It contains
g0 , of course. Finally, we choose H ⊆ N in such a way that H /C is an -Sylow
subgroup of N/C which contains the -subgroup g0 /C. By construction H is
-hyper-elementary.
In the next step we study the cardinality of the set

{gH ∈ G/H : g0 gH = gH } = gH ∈ G/H : g −1 g0 g ∈ H .

Suppose that g −1 g0 g ∈ H . Then g −1 Cg ⊆ g −1 g0 g = g −1 g0 g ⊆ H . But, the
two sides having coprime orders, the projection map g −1 Cg −→ H /C has to be
the trivial map. It follows that g −1 Cg = C which means that g ∈ N . This shows
that
{gH ∈ G/H : g0 gH = gH } = {gH ∈ N/H : g0 gH = gH }.
The cardinality of the right-hand side is the number of g0 -orbits in N/H which
consist of one point only. We note that the subgroup C, being normal in N and
contained in H , acts trivially on N/H . Hence the g0 -orbits coincide with the
orbits of the -group g0 /C. But, quite generally, the cardinality of an orbit, being
the index of the stabilizer of any point in the orbit, divides the order of the acting
group. It follows that the cardinality of any g0 -orbit in N/H is a power of . We
conclude that

{gH ∈ N/H : g0 gH = gH } ≡ |N/H | = [N : H ] mod .

But by the choice of H we have [N : H ]. This establishes our claim.
By (2.4.2) we now may choose an element zg ∈ PF (G, Hhe ), for any g ∈ G, such
that tg (zg ) = 1. We then have

tg (zg − 1) = 0 for any g ∈ G,
g ∈G

2.5 Clifford Theory 67

and (2.4.1) implies that
(zg − 1) = 0.
g ∈G

Multiplying out the left-hand side and using that PF (G, Hhe ) is additively and mul-
tiplicatively closed easily shows that 1 ∈ PF (G, Hhe ).

2.5 Clifford Theory

As before F is an arbitrary field. We fix a normal subgroup N in our finite group G.
Let W be an F [N]-module. It is given by a homomorphism of F -algebras

π: F [N ] −→ EndF (W ).

For any g ∈ G we now define a new F [N]-module g ∗ (W ) by the composite homo-
morphism
π
F [N] −→ F [N ] − EndF (W )
→
h −→ ghg −1
or equivalently by

F [N ] × g ∗ (W ) −→ g ∗ (W )
(h, w) −→ ghg −1 w.

Remark 2.5.1
i. dimF g ∗ (W ) = dimF W .
ii. The map U → g ∗ (U ) is a bijection between the set of F [N ]-submodules of W
and the set of F [N]-submodules of g ∗ (W ).
iii. W is simple if and only if g ∗ (W ) is simple.
∗ ∗
iv. g1 (g2 (W )) = (g2 g1 )∗ (W ) for any g1 , g2 ∈ G.
v. Any F [N]-module homomorphism α : W1 −→ W2 also is a homomorphism of
F [N ]-modules α : g ∗ (W1 ) −→ g ∗ (W2 ).

Proof Trivial or straightforward.

Suppose that g ∈ N . One checks that then
∼
=
W −→ g ∗ (W )
w −→ gw

is an isomorphism of F [N ]-modules. Together with Remark 2.5.1.iv/v this implies
that


G/N × (MF [N ] / ∼) −→ MF [N ] / ∼
= =
gN, {W } −→ g ∗ (W )
∼
is a well-deﬁned action of the group G/N on the set MF [N ] / =. By Remark 2.5.1.iii
this action respects the subset F [N ]. For any {W } in F [N ] we put

IG (W ) := g ∈ G : g ∗ (W ) = {W } .

As a consequence of Remark 2.5.1.iv this is a subgroup of G, which contains N of
course.

Remark 2.5.2 Let V be an F [G]-module, and let g ∈ G be any element; then the
map

set of all F [N ]- set of all F [N ]-
−→
submodules of V submodules of V
W −→ gW

is an inclusion preserving bijection; moreover, for any F [N ]-submodule W ⊆ V we
have:
i. The map
∼
=
g ∗ (W ) −→ g −1 W
w −→ g −1 w

is an isomorphism of F [N]-modules;
ii. gW is a simple F [N]-module if and only if W is a simple F [N ]-module;
iii. if W1 ∼ W2 are isomorphic F [N ]-submodules of V then also gW1 ∼ gW2 are
= =
isomorphic as F [N]-modules.

Proof For h ∈ N we have

h(gW ) = g g −1 hg W = gW

since g −1 hg ∈ N . Hence gW indeed is an F [N]-submodule, and the map in the
assertion is well deﬁned. It obviously is inclusion preserving. Its bijectivity is im-
mediate from g −1 (gW ) = W = g(g −1 W ). The assertion ii. is a direct consequence.
The map in i. clearly is an F -linear isomorphism. Because of

g −1 ghg −1 w = h g −1 w for any h ∈ N and w ∈ W

it is an F [N]-module isomorphism. The last assertion iii. follows from i. and Re-
mark 2.5.1.v.

2.5 Clifford Theory 69

Theorem 2.5.3 (Clifford) Let V be a simple F [G]-module; we then have:
i. V is semisimple as an F [N ]-module;
˜
ii. let W ⊆ V be a simple F [N]-submodule, and let W ⊆ V be the {W }-isotypic
component; then
˜
a. W is a simple F [IG (W )]-module, and
b. V ∼ IndG (W ) (W ) as F [G]-modules.
= IG
˜

Proof Since V is of finite length as an F [N ]-module we find a simple F [N ]-
submodule W ⊆ V . Then gW , for any g ∈ G, is another simple F [N ]-submodule
by Remark 2.5.2.ii. Therefore V0 := g∈G gW is, on the one hand, a semisimple
F [N ]-module by Proposition 1.1.4. On the other hand it is, by definition, a nonzero
F [G]-submodule of V . Since V is simple we must have V0 = V , which proves the
˜
assertion i. As an F [N ]-submodule the {W }-isotypic component W of V is of the
form W = W1 ⊕ · · · ⊕ Wm with simple F [N ]-submodules Wi ∼ W . Let first g be
˜ =
an element in IG (W ). Using Remark 2.5.2.i/iii we obtain gWi ∼ gW ∼ W for any
= =
˜ ˜ ˜
1 ≤ i ≤ m. It follows that gWi ⊆ W for any 1 ≤ i ≤ m and hence g W ⊆ W . We see
˜
that W is an F [IG (W )]-submodule of V . For a general g ∈ G we conclude from
˜
Remark 2.5.2 that g W is the {gW }-isotypic component of V . We certainly have

V= ˜
gW .
g∈G/IG (W )

˜ ˜
Two such submodules g1 W and g2 W , being isotypic components, either are equal
˜ = g2 W then g −1 g1 W = W , hence g −1 g1 W ∼ W ,
or have zero intersection. If g1 W ˜ ˜ ˜ =
2 2
−1
and therefore g2 g1 ∈ IG (W ). We see that in fact

V= ˜
gW . (2.5.1)
g∈G/IG (W )

˜
The inclusion W ⊆ V induces, by the first Frobenius reciprocity, the F [G]-module
homomorphism

IndG (W ) (W ) ∼ F [G] ⊗F [IG (W )] W −→ V
IG
˜ = ˜

ag g ⊗ w −→
˜ ˜
ag g w.
g∈G g∈G

Since V is simple it must be surjective. But both sides have the same dimension
˜
as F -vector spaces [G : IG (W )] · dimF W , the left-hand side by the argument in
the proof of Remark 2.3.2 and the right-hand side by (2.5.1). Hence this map is an
isomorphism which proves ii.b. Finally, since F [G] ⊗F [IG (W )] W ∼ V is a sim-
˜ =
ple F [G]-module it follows from Remark 2.3.3 that W ˜ is a simple F [IG (W )]-
module.

In the next section we will need the following particular consequence of this
result. But first we point out that for an F [N ]-module W of dimension dimF W = 1


the describing algebra homomorphism π is of the form

π: F [N ] −→ F.

The corresponding homomorphism for g ∗ (W ) then is x → π(gxg −1 ). Since an
endomorphism of a one-dimensional F -vector space is given by multiplication by a
scalar we have g ∈ IG (W ) if and only if there is a scalar a ∈ F × such that

aπ(h)w = π ghg −1 aw for any h ∈ N and w ∈ W.

It follows that

IG (W ) = g ∈ G : π ghg −1 = π(h) for any h ∈ N (2.5.2)

if dimF W = 1.

Remark 2.5.4 Suppose that N is abelian and that F is a splitting field for N ; then
any simple F [N ]-module W has dimension dimF W = 1.

Proof Let π : F [N] −→ EndF (W ) be the algebra homomorphism describing W .
Since N is abelian we have im(π) ⊆ EndF [N ] (W ). By our assumption on the
field F the latter is equal to F . This means that any element in F [N ] acts on
W by multiplication by a scalar in F . Since W is simple this forces it to be one-
dimensional.

Proposition 2.5.5 Let H be an -hyper-elementary group with cyclic normal sub-
group C such that |C| and H /C is an -group, and let V be a simple F [H ]-
module; we suppose that
a. F is a splitting field for C,
b. V does not contain the trivial F [C]-module, and
c. the subgroup C0 := {c ∈ C : cg = gc for any g ∈ H } acts trivially on V ;
then there exists a proper subgroup H H and an F [H ]-module V such that
∼
V = IndH (V ) as F [H ]-modules.
H

Proof We pick any simple F [C]-submodule W ⊆ V . By applying Clifford’s Theo-
rem 2.5.3 to the normal subgroup C and the module W it suffices to show that

IH (W ) = H.

According to Remark 2.5.4 the module W is one-dimensional and given by an alge-
bra homomorphism π : F [C] −→ F , and by (2.5.2) we have

IH (W ) = g ∈ H : π gcg −1 = π(c) for any c ∈ C .

The assumption c. means that

C0 ⊆ C1 := ker(π|C).

2.6 Brauer’s Induction Theorem 71

We immediately note that any subgroup of the cyclic normal subgroup C also is
normal in H . By assumption b. we find an element c2 ∈ C C1 so that π(c2 ) = 1.
Let L ⊆ H be an -Sylow subgroup. We claim that we find an element g0 ∈ L such
−1
that π(g0 c2 g0 ) = π(c2 ). Then g0 ∈ IH (W ) which establishes what we wanted. We
point out that, since H = C × L as sets, we have

C0 = {c ∈ C : cg = gc for any g ∈ L}.

Arguing by contradiction we assume that π(gc2 g −1 ) = π(c2 ) for any g ∈ L. Then

gc2 C1 g −1 = gc2 g −1 C1 = c2 C1 for any g ∈ L.

This means that we may consider c2 C1 as an L-set with respect to the conjugation
action. Since L is an -group the cardinality of any L-orbit in c2 C1 is a power of .
On the other hand we have |C1 | = |c2 C1 |. There must therefore exist an element
c0 ∈ c2 C1 such that
gc0 g −1 = c0 for any g ∈ L
(i.e. an L-orbit of cardinality one). We conclude that c0 ∈ C0 ⊆ C1 and hence
c2 C1 = c0 C1 = C1 . This is in contradiction to c2 ∈ C1 .

2.6 Brauer’s Induction Theorem
In this section F is a field of characteristic zero, and G continues to be any finite
group.

Definition
i. Let be a prime number. A finite group H is called -elementary if it is a direct
product H = C × L of a cyclic group C and an -group L.
ii. A finite group is called elementary if it is -elementary for some prime number .

Exercise
i. If H is -elementary then H = C × L is the direct product of a cyclic group C
of order prime to and an -group L.
ii. Any -elementary group is -hyper-elementary.
iii. Any subgroup of an -elementary group is -elementary.

Let He denote the family of elementary subgroups of G.

Theorem 2.6.1 (Brauer) Suppose that F is a splitting field for every subgroup of G;
then
indG RF (H ) = RF (G).
H
H ∈He


Proof By Corollary 2.3.5 each indG (RF (H )) is an ideal in the ring RF (G). Hence
H
the left-hand side of the asserted identity is an ideal in the right-hand side. To obtain
equality we therefore need only to show that the unit element 1G ∈ RF (G) lies in the
left-hand side. According to Solomon’s Theorem 2.4.8 together with Lemma 2.4.6.i
we have

1G ∈ Z F [G/H ] = Zb [G/H ] = Z indG (1H )
H
H ∈Hhe H ∈Hhe H ∈Hhe

⊆ indG RF (H ) .
H
H ∈Hhe

By the transitivity of induction this reduces us to the case that G is -hyper-
elementary for some prime number . We now proceed by induction with respect to
the order of G and assume that our assertion holds for all proper subgroups H G.
We also may assume, of course, that G is not elementary. Using the transitivity of
induction again it then suffices to show that

1G ∈ indG RF H
H .
H G

Let C ⊆ G be the cyclic normal subgroup of order prime to such that G/C is an
-group. We fix an -Sylow subgroup L ⊆ G. Then G = C × L as sets. In C we
have the (cyclic) subgroup

C0 := {c ∈ C : cg = gc for any g ∈ G}.

Then
H0 := C0 × L
is an -elementary subgroup of G. Since G is not elementary we must have H0 G.
We consider the induction IndG0 (F ) of the trivial F [H0 ]-module F . By semisim-
H
plicity it decomposes into a direct sum

IndG0 (F ) = V0 ⊕ V1 ⊕ · · · ⊕ Vr
H

of simple F [G]-modules Vi . We recall that IndG0 (F ) is the F -vector space of all
H
functions φ : G/H0 −→ F with the G-action given by
g
φ g H0 = φ g −1 g H0 .

This G-action fixes a function φ if and only if φ(gg H0 ) = φ(g H0 ) for any g, g ∈
G, i.e. if and only if φ is constant. It follows that the one-dimensional subspace
of constant functions is the only simple F [G]-submodule in IndG0 (F ) which is
H
isomorphic to the trivial module. We may assume that V0 is this trivial submodule.
In RF (G) we then have the equation

1G = indG0 (1H0 ) − [V1 ] − · · · − [Vr ].
H

2.6 Brauer’s Induction Theorem 73

This reduces us further to showing that, for any 1 ≤ i ≤ r, we have

[Vi ] ∈ indGi RF (Hi )
H

for some proper subgroup Hi G. This will be achieved by applying the criterion in
Proposition 2.5.5. By our assumption on F it remains to verify the conditions b. and
c. in that proposition for each V1 , . . . , Vr . Since C0 is central in G and is contained
in H0 it acts trivially on IndG0 (F ) and a fortiori on any Vi . This is condition c. For
H
b. we note that CH0 = G and C ∩ H0 = C0 . Hence the inclusion C ⊆ G induces a
bijection C/C0 −→ G/H0 . It follows that the map
∼
=
IndG0 (F ) −→ IndC0 (F )
H C

φ −→ φ|(C/C0 )

is an isomorphism of F [C]-modules. It maps constant functions to constant
functions and hence the unique trivial F [G]-submodule V0 to the unique trivial
F [C]-submodule. Therefore Vi , for 1 ≤ i ≤ r, cannot contain any trivial F [C]-
submodule.

Lemma 2.6.2 Let H be an elementary group, and let N0 ⊆ H be a normal sub-
group such that H /N0 is not abelian; then there exists a normal subgroup N0 ⊆
N ⊆ H such that N/N0 is abelian but is not contained in the center Z(H /N0 ) of
H /N0 .

Proof With H also H /N0 is elementary (if H = C × L with C and L having co-
prime orders then H /N0 ∼ C/C ∩ N0 × L/L ∩ N0 ). We therefore may assume
=
without loss of generality that N0 = {1}. Step 1: We assume that H is an -group
for some prime number . By assumption we have Z(H ) = H so that H /Z(H ) is
an -group ={1}. We pick a cyclic normal subgroup {1} = N = g ⊆ H /Z(H ).
Let Z(H ) ⊆ N ⊆ H be the normal subgroup such that N/Z(H ) = N and let g ∈ N
be a preimage of g. Clearly N = Z(H ), g is abelian. But Z(H ) N since g = 1.
Step 2: In general let H = C × L where C is cyclic and L is an -group. With H also
L is not abelian. Applying Step 1 to L we ﬁnd a normal abelian subgroup NL ⊆ L
such that NL Z(L). Then N := C × NL is a normal abelian subgroup of H such
that N Z(H ) = C × Z(L).

Lemma 2.6.3 Let H be an elementary group, and let W be a simple F [H ]-module;
we suppose that F is a splitting ﬁeld for all subgroups of H ; then there exists a
subgroup H ⊆ H and a one-dimensional F [H ]-module W such that

W ∼ IndH W
= H

as F [H ]-modules.


Proof We choose H ⊆ H to be a minimal subgroup (possibly equal to H ) such
that there exists an F [H ]-module W with W ∼ IndH (W ), and we observe that
= H
W necessarily is a simple F [H ]-module by Remark 2.3.3. Let

π : H −→ EndF W

be the corresponding algebra homomorphism, and put N0 := ker(π ). We claim
that H /N0 is abelian. Suppose otherwise. Then, by Lemma 2.6.2, there exists a
normal subgroup N0 ⊆ N ⊆ H such that N/N0 is abelian but is not contained in
¯
Z(H /N0 ). Let W ⊆ W be a simple F [N]-submodule. By Clifford’s Theorem 2.5.3
we have
W ∼ IndH
= ˜
¯
I H
¯ (W )
(W )

˜
¯ ¯
where W denotes the {W }-isotypic component of W . Transitivity of induction im-
plies

W ∼ IndH IndH
= ˜
¯ ∼ IndH
= ˜
¯
H I H
¯ (W )
(W ) I H
¯ (W ).
(W )

¯
By the minimality of H we therefore must have IH (W ) = H which means that
W =W ˜ is {W }-isotypic.
¯ ¯
¯
On the other hand, W is an F [H /N0 ]-module. Hence W is a simple F [N/N0 ]-
module for the abelian group N/N0 . Remark 2.5.4 then implies (note that
¯ ¯ ¯
EndF [N/N0 ] (W ) = EndF [N ] (W ) = F ) that W is one-dimensional given by an al-
gebra homomorphism
χ: F [N/N0 ] −→ F.
¯
It follows that any h ∈ N acts on the {W }-isotypic module W by multiplication by
the scalar χ(hN0 ). In other words the injective homomorphism

H /N0 −→ EndF W
hN0 −→ π (h)

satisﬁes
π (h) = χ(hN0 ) · idW for any h ∈ N.
But χ(hN0 ) · idW lies in the center of EndF (W ). The injectivity of the homomor-
phism therefore implies that N/N0 lies in the center of H /N0 . This is a contradic-
tion.
We thus have established that H /N0 is abelian. Applying Remark 2.5.4 to W
viewed as a simple F [H /N0 ]-module we conclude that W is one-dimensional.

Theorem 2.6.4 (Brauer) Suppose that F is a splitting ﬁeld for any subgroup of G,
and let x ∈ RF (G) be any element; then there exist integers m1 , . . . , mr , elementary

2.7 Splitting Fields 75

subgroups H1 , . . . , Hr , and one-dimensional F [Hi ]-modules Wi such that
r
x= mi indGi [Wi ] .
H
i=1

Proof Combine Theorem 2.6.1, Proposition 1.7.1, Lemma 2.6.3, and the transitivity
of induction.

2.7 Splitting Fields

Again F is a field of characteristic zero.

Lemma 2.7.1 Let E/F be any extension field, and let V and W be two finitely
generated F [G]-modules; we then have

HomE[G] (E ⊗F V , E ⊗F W ) = E ⊗F HomF [G] (V , W ).

Proof First of all we observe, by comparing dimensions, that

HomE (E ⊗F V , E ⊗F W ) = E ⊗F HomF (V , W )

holds true. We now consider U := HomF (V , W ) as an F [G]-module via

G × U −→ U
(g, f ) −→ g f := gf g −1 .

Then HomF [G] (V , W ) = U G := {f ∈ U : g f = f for any g ∈ G} is the {F }-
isotypic component of U for the trivial F [G]-module F . Correspondingly we obtain

HomE[G] (E ⊗F V , E ⊗F W ) = HomE (E ⊗F V , E ⊗F W )G = (E ⊗F U )G .

This reduces us to proving that

(E ⊗F U )G = E ⊗F U G

for any F [G]-module U . The element

1
εG := g ∈ F [G] ⊆ E[G]
|G|
g∈G

is an idempotent with the property that

U G = εG · U,


and hence

(E ⊗F U )G = εG · (E ⊗F U ) = E ⊗F εG · U = E ⊗F U G .

Theorem 2.7.2 (Brauer) Let e be the exponent of G, and suppose that F contains
a primitive eth root of unity; then F is a splitting field for any subgroup of G.

¯
Proof We fix an algebraic closure F of F . Step 1: We show that, for any finitely
¯ [G]-module V , there is an F [G]-module V such that
generated F ¯

¯∼ ¯
V = F ⊗F V ¯
as F [G]-modules.

According to Brauer’s Theorem 2.6.4 we find integers m1 , . . . , mr , subgroups
¯ ¯
H1 , . . . , Hr of G, and one-dimensional F [Hi ]-modules Wi such that
r
¯
[V ] = ¯ ¯
mi F [G] ⊗F [Hi ] Wi
¯
i=1

= ¯ ¯ m
F [G] ⊗F [Hi ] Wi i −
¯ ¯ ¯ −m
F [G] ⊗F [Hi ] Wi i
¯
mi >0 mi <0

= ¯ ¯ m
¯ − ¯ ¯ −m
¯ .
mi >0 mi <0

¯ ¯ ¯ ¯
Let πi : F [Hi ] −→ F denote the F -algebra homomorphism describing Wi . We have
πi (h)e = π (he ) = π (1) = 1 for any h ∈ H . Our assumption on F therefore implies
i i i
that πi (F [Hi ]) ⊆ F . Hence the restriction πi |F [Hi ] describes a one-dimensional
F [Hi ]-module Wi such that

W i ∼ F ⊗F W i
¯ = ¯ ¯
as F [Hi ]-modules.

We define the F [G]-modules

V+ := F [G] ⊗F [Hi ] Wimi and V− := F [G] ⊗F [Hi ] Wi−mi .
mi >0 mi <0

Then

¯
F ⊗F V+ = ¯
F ⊗F F [G] ⊗F [Hi ] Wimi = ¯
F [G] ⊗F [Hi ] Wimi
mi >0 mi >0

= ¯ ¯
F [G] ⊗F [Hi ] F [Hi ] ⊗F [Hi ] Wimi
¯
mi >0

2.7 Splitting Fields 77

= ¯ ¯
F [G] ⊗F [Hi ] F ⊗F Wimi
¯
mi >0

∼
= ¯ ¯ m
¯
mi >0

and similarly
F ⊗F V− ∼
¯ = ¯ ¯ −m
F [G] ⊗F [Hi ] Wi i .
¯
mi <0

It follows that
¯ ¯ ¯
[V ] = [F ⊗F V+ ] − [F ⊗F V− ]
or, equivalently, that
¯ ¯ ¯
V ⊕ (F ⊗F V− ) = [F ⊗F V+ ].

Using Corollary 2.3.7.i/iii we deduce that

V ⊕ (F ⊗F V− ) ∼ F ⊗F V+
¯ ¯ = ¯ ¯
as F [G]-modules.

If V− is nonzero then V− = U ⊕ V− is a direct sum of F [G]-modules where
U is simple. On the other hand let V+ = U1 ⊕ · · · ⊕ Um be a decomposition
¯ ¯
into simple F [G]-modules. Then F ⊗F U is a direct summand of F ⊗F V− and
hence is isomorphic to a direct summand of F ¯ ⊗F V+ . We therefore must have
¯ ¯
HomF [G] (F ⊗F U, F ⊗F Uj ) = {0} for some 1 ≤ j ≤ m. Lemma 2.7.1 implies
¯
that HomF [G] (U, Uj ) = {0}. Hence U ∼ Uj as F [G]-modules. We conclude that
=
V+ ∼ U ⊕ V+ with V+ := i=j Ui , and we obtain
=

V ⊕ F ⊗F V− ⊕ (F ⊗F U ) ∼ F ⊗F V+ ⊕ (F ⊗F U ).
¯ ¯ ¯ = ¯ ¯

The Jordan–Hölder Proposition 1.1.2 then implies that

V ⊕ F ⊗F V− ∼ F ⊗F V+ .
¯ ¯ = ¯

By repeating this argument we arrive after ﬁnitely many steps at an F [G]-module
V such that
V ∼ F ⊗F V .
¯= ¯

Step 2: Let now V be a simple F [G]-module, and let F ⊗F V = V1 ⊕ · · · ⊕ V¯m be
¯ ¯
a decomposition into simple F¯ [G]-modules. By Step 1 we ﬁnd F [G]-modules Vi
such that
Vi ∼ F ⊗F Vi .
¯ = ¯

For any 1 ≤ i ≤ m, the F [G]-module Vi necessarily is simple, and we have
¯ ¯
HomF [G] (F ⊗F Vi , F ⊗F V ) = {0}. Hence HomF [G] (Vi , V ) = {0} by Lemma 2.7.1.
¯
∼ V for any 1 ≤ i ≤ m. By comparing dimensions we conclude
It follows that Vi =


¯ ¯
that m = 1. This means that F ⊗F V is a simple F [G]-module. Using Lemma 2.7.1
again we obtain
¯ ¯ ¯
F ⊗F EndF [G] (V ) = EndF [G] (F ⊗F V ) = F .
¯

Hence EndF [G] (V ) = F must be one-dimensional. This shows that F is a splitting
field for G. Step 3: Let H ⊆ G be any subgroup with exponent eH . Since eH divides
e the field F a fortiori contains a primitive eH th root of unity. Hence F also is a
splitting field for H .

2.8 Properties of the Cartan–Brauer Triangle
We go back to the setting from the beginning of this chapter: k is an algebraically
closed field of characteristic p > 0, R is a (0, p)-ring for k with maximal ideal
mR = RπR , and K denotes the field of fractions of R.
The ring R will be called splitting for our finite group G if K contains a primitive
eth root of unity ζ where e is the exponent of G. By Theorem 2.7.2 the field K, in
this case, is a splitting field for any subgroup of G. This additional condition can
easily be achieved by defining K := K(ζ ) and R := {a ∈ K : NormK /K (a) ∈ R};
then R is a (0, p)-ring for k which is splitting for G.
It is our goal in this section to establish the deeper properties of the Cartan–
Brauer triangle

dG
RK (G) Rk (G)

eG cG

K0 (k[G]).

Lemma 2.8.1 For any subgroup H ⊆ G the diagram

dH
RK (H ) Rk (H )

indG
H indG
H
dG
RK (G) Rk (G)

is commutative.

Proof Let W be a finitely generated K[H ]-module. We choose a lattice L ⊆ W
which is H -invariant. Then

indG dH [W ]
H = indG [L/πR L] = k[G] ⊗k[H ] (L/πR L) .
H

2.8 Properties of the Cartan–Brauer Triangle 79

Moreover, R[G] ⊗R[H ] L ∼ L[G:H ] is a G-invariant lattice in K[G] ⊗K[H ] W ∼
= =
W [G:H ] (compare the proof of Remark 2.3.2). Hence

dG indG [W ]
H = dG K[G] ⊗K[H ] W
= R[G] ⊗R[H ] L /πR R[G] ⊗R[H ] L
= k[G] ⊗k[H ] (L/πR L) .

Lemma 2.8.2 We have

Rk (G) = indG Rk (H ) .
H
H ∈He

Proof Since the indG (Rk (H )) are ideals in Rk (G) it suffices to show that the unit
H
element 1k[G] ∈ Rk (G) lies in the right-hand side. We choose R to be splitting for G.
By Brauer’s induction Theorem 2.6.1 we have

1K[G] ∈ indG RK (H )
H
H ∈He

where 1K[G] is the unit element in RK (G). Using Lemma 2.8.1 we obtain

dG (1K[G] ) ∈ dG indG RK (H )
H = indG dH RK (H )
H
H ∈He H ∈He

⊆ indG Rk (H ) .
H
H ∈He

It is trivial to see that dG (1K[G] ) = 1k[G] .

Theorem 2.8.3 The decomposition homomorphism

dG : RK (G) −→ Rk (G)

is surjective.

Proof By Lemma 2.8.1 we have

dG RK (G) ⊇ dG indG RK (H )
H = indG dH RK (H ) .
H
H ∈He H ∈He

Because of Lemma 2.8.2 it therefore suffices to show that dH (RK (H )) = Rk (H ) for
any H ∈ He . This means we are reduced to proving our assertion in the case where
the group G is elementary. Then G = H × P is the direct product of a group H of
order prime to p and a p-group P . By Proposition 1.7.1 it suffices to show that the
class [W ] ∈ Rk (G), for any simple k[G]-module W , lies in the image of dG . Viewed


as a k[P ]-module W must contain the trivial k[P ]-module by Proposition 2.2.7. We
deduce that
W P := {w ∈ W : gw = w for any g ∈ P } = {0}.
Since P is a normal subgroup of G the k[P ]-submodule W P in fact is a k[G]-
submodule of W . But W is simple. Hence W P = W which means that k[G] acts
on W through the projection map k[G] −→ k[H ]. According to Corollary 2.2.6
we find a simple K[H ]-module V together with a G-invariant lattice L ⊆ V such
that L/πR L ∼ W as k[H ]-modules. Viewing V as a K[G]-module through the
=
projection map K[G] −→ K[H ] we obtain [V ] ∈ RK (G) and dG ([V ]) = [W ].

Theorem 2.8.4 Let p m be the largest power of p which divides the order of G; the
Cartan homomorphism cG : K0 (k[G]) −→ Rk (G) is injective, its cokernel is finite,
and p m Rk (G) ⊆ im(cG ).

Proof Step 1: We show that p m Rk (G) ⊆ im(cG ) holds true. It is trivial from the
definition of the Cartan homomorphism that, for any subgroup H ⊆ G, the diagram

cH
K0 (k[H ]) Rk (H )

[P ]−→[k[G]⊗k[H ] P ] indG
H
cG
K0 (k[G]) Rk (G)

is commutative. It follows that
indG im(cH ) ⊆ im(cG ).
H

Lemma 2.8.2 therefore reduces us to the case that G is an elementary group. Let W
be any simple k[G]-module. With the notations of the proof of Theorem 2.8.3 we
have seen there that k[G] acts on W through the projection map k[G] −→ k[H ].
Viewed as a k[H ]-module W is projective by Remark 1.7.3. Hence k[G] ⊗k[H ] W
is a finitely generated projective k[G]-module. We claim that
cG k[G] ⊗k[H ] W = |G/H | · [W ]
holds true. Using the above commutative diagram as well as Proposition 2.3.4 we
obtain
cG k[G] ⊗k[H ] W = indG [W ] = k[G] ⊗k[H ] k · [W ].
H
In order to analyze the k[G]-module k[G] ⊗k[H ] k let h, h ∈ H , g ∈ P , and a ∈ k.
Then
h gh ⊗ a = ghh ⊗ a = g ⊗ a = gh ⊗ a.
This shows that H acts trivially on k[G] ⊗k[H ] k. In other words, k[G] acts on
k[G] ⊗k[H ] k through the projection map k[G] −→ k[P ]. It then follows from
Proposition 2.2.7 that all simple subquotients in a composition series of the k[G]-


module k[G] ⊗k[H ] k are trivial k[G]-modules. We conclude that

k[G] ⊗k[H ] k = dimk k[G] ⊗k[H ] k · 1 = |G/H | · 1

(where 1 ∈ Rk (G) is the unit element).
Step 2: We know from Proposition 1.7.1 that, as an abelian group, Rk (G) ∼ Zr
=
for some r ≥ 1. It therefore follows from Step 1 that Rk (G)/ im(cG ) is isomorphic
to a factor group of the finite group Zr /p m Zr .
Step 3: It is a consequence of Proposition 1.7.4 that K0 (k[G]) and Rk (G) are
isomorphic to Zr for the same integer r ≥ 1. Hence

id ⊗cG : Q ⊗Z K0 k[G] −→ Q ⊗Z Rk (G)

is a linear map between two Q-vector spaces of the same finite dimension r. Its
injectivity is equivalent to its surjectivity. Let a ∈ Q and x ∈ Rk (G). By Step 1 we
find an element y ∈ K0 (k[G]) such that cG (y) = p m x. Then

a a
(id ⊗cG ) m
⊗ y = m ⊗ p m x = a ⊗ x.
p p

This shows that id ⊗cG and consequently cG are injective.

In order to discuss the third homomorphism eG we first introduce two bilinear
forms. We start from the maps

(MK[G] / ∼ × (MK[G] / ∼ −→ Z
=) =)
{V }, {W } −→ dimK HomK[G] (V , W )

and

(Mk[G] / ∼) × (Mk[G] / ∼) −→ Z
= =
{P }, {V } −→ dimk Homk[G] (P , V ).

They extend to Z-bilinear maps

Z[MK[G] ] × Z[MK[G] ] −→ Z

and
Z[Mk[G] ] × Z[Mk[G] ] −→ Z.
Since K[G] is semisimple we have, for any exact sequence 0 → V1 → V → V2 → 0
in MK[G] , that V ∼ V1 ⊕ V2 and hence that
=

dimK HomK[G] (V , W ) − dimK HomK[G] (V1 , W ) − dimK HomK[G] (V2 , W ) = 0.


The corresponding fact in the “variable” W holds as well, of course. The first map
therefore induces a well-defined Z-bilinear form

, K[G] : RK (G) × RK (G) −→ Z
[V ], [W ] −→ dimK HomK[G] (V , W ).

Even though k[G] might not be semisimple any exact sequence 0 → P1 → P →
∼
P2 → 0 in Mk[G] still satisfies P = P1 ⊕ P2 as a consequence of Lemma 1.6.2.ii.
Hence we again have

dimk Homk[G] (P , V ) − dimk Homk[G] (P1 , V ) − dimk Homk[G] (P2 , V ) = 0

for any V in Mk[G] . Furthermore, for any P in Mk[G] and any exact sequence
0 → V1 → V → V2 → 0 in Mk[G] we have, by the definition of projective modules,
the exact sequence

0 −→ Homk[G] (P , V1 ) −→ Homk[G] (P , V ) −→ Homk[G] (P , V2 ) −→ 0.

Hence once more

dimk Homk[G] (P , V ) − dimk Homk[G] (P , V1 ) − dimk Homk[G] (P , V2 ) = 0.

This shows that the second map also induces a well-defined Z-bilinear form

, k[G] : K0 k[G] × Rk (G) −→ Z
[P ], [V ] −→ dimk Homk[G] (P , V ).

If {V1 }, . . . , {Vr } are the isomorphism classes of the simple K[G]-modules then
[V1 ], . . . , [Vr ] is a Z-basis of RK (G) by Proposition 1.7.1. We have

dimK EndK[G] (Vi ) if i = j,
[Vi ], [Vj ] K[G] =
0 if i = j.

In particular, if K is a splitting field for G then

1 if i = j,
[Vi ], [Vj ] K[G] =
0 if i = j.

Let {P1 }, . . . , {Pt } be the isomorphism classes of finitely generated indecomposable
projective k[G]-modules. By Proposition 1.7.4.iii the [P1 ], . . . , [Pt ] form a Z-basis
of K0 (k[G]), and the [P1 / Jac(k[G])P1 ], . . . , [Pt / Jac(k[G])Pt ] form a Z-basis of
Rk (G) by Proposition 1.7.4.i. We have


Homk[G] Pi , Pj / Jac k[G] Pj = Homk[G] Pi / Jac k[G] Pi , Pj / Jac k[G] Pj

Endk[G] (Pi / Jac(k[G])Pi ) if i = j,
=
{0} if i = j

k if i = j,
=
{0} if i = j,

where the latter identity comes from the fact that the algebraically closed field k is
a splitting field for G. Hence

1 if i = j,
[Pi ], Pj / Jac k[G] Pj =
k[G] 0 if i = j.

Exercise 2.8.5
i. If K is a splitting field for G then the map

∼
=
RK (G) −→ HomZ RK (G), Z
x −→ x, . K[G]

is an isomorphism of abelian groups.
ii. The maps

∼
= ∼
=
K0 k[G] −→ HomZ Rk (G), Z and Rk (G) −→ HomZ K0 k[G] , Z
y −→ y, . k[G] z −→ . , z k[G]

are isomorphisms of abelian groups.

Lemma 2.8.6 We have

y, dG (x) k[G] = eG (y), x K[G]

for any y ∈ K0 (k[G]) and x ∈ RK (G).

Proof It suffices to consider elements of the form y = [P /πR P ] for some finitely
generated projective R[G]-module P (see Proposition 2.1.1) and x = [V ] for some
finitely generated K[G]-module V . We pick a G-invariant lattice L ⊆ V . The as-
serted identity then reads

dimk Homk[G] (P /πR P , L/πR L) = dimK HomK[G] (K ⊗R P , V ).


We have

Homk[G] (P /πR P , L/πR L) = HomR[G] (P , L/πR L)
= HomR[G] (P , L)/ HomR[G] (P , πR L)
= HomR[G] (P , L)/πR HomR[G] (P , L)
= k ⊗R HomR[G] (P , L) (2.8.1)

where the second identity comes from the projectivity of P as an R[G]-module. On
the other hand

HomK[G] (K ⊗R P , V ) = HomR[G] (P , V )

−i
= HomR[G] P , πR L
i≥0

−i
= HomR[G] P , πR L
i≥0

−i
= πR HomR[G] (P , L)
i≥0

= K ⊗R HomR[G] (P , L). (2.8.2)

For the third identity one has to observe that, since P is finitely generated as an
−i
R-module, any R-module homomorphism P −→ V = i≥0 πR L has to have its
−i
image inside πR L for some sufficiently large i.
Both, P being a direct summand of some R[G]m (Remark 2.1.2) and L by defini-
tion are free R-modules. Hence HomR (P , L) is a finitely generated free R-module.
The ring R being noetherian the R-submodule HomR[G] (P , L) is finitely generated
as well. Lemma 2.2.1.i then implies that HomR[G] (P , L) ∼ R s is a free R-module.
=
We now deduce from (2.8.1) and (2.8.2) that

Homk[G] (P /πR P , L/πR L) ∼ k s
= and HomK[G] (K ⊗R P , V ) ∼ K s ,
=

respectively.

Theorem 2.8.7 The homomorphism eG : K0 (k[G]) −→ RK (G) is injective and its
image is a direct summand of RK (G).

Proof Step 1: We assume that R is splitting for G. By Theorem 2.8.3 the map
dG : RK (G) −→ Rk (G) is surjective. Since, by Proposition 1.7.1, Rk (G) is a free
abelian group we find a homomorphism s : Rk (G) −→ RK (G) such that dG ◦s = id.
It follows that

Hom(s, Z) ◦ Hom(dG , Z) = Hom(dG ◦ s, Z) = Hom(id, Z) = id .

The Cartan–Brauer Triangle in Modular Representation Theory

The Cartan–Brauer Triangle in Modular Representation Theory

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