Group Theory

GROUPS
ADVANCED GROUP THEORY
References
GROUP THEORY
Shinoj K.M.
Department of Mathematics
St.Joseph’s College,Devagiri
Kozhikode-8
shinusaraswathy@gmail.com
8 December 2018
Shinoj K.M. Department of MathematicsGROUP THEORY

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Outline
1 GROUPS
Order
Cyclic Groups
Normal Subgroups
2 ADVANCED GROUP THEORY
Direct product of groups
Fundamental Theorem of ﬁnite Abelian groups
Classiﬁcation of groups of order up to 10

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References
Order
Cyclic Groups
Normal Subgroups
Deﬁnition
A set G together with a binary operation ∗ is called a group if
G is closed w.r.t. ∗
∗ is associative
there exists an identity element in G
for each element in G there is an inverse in G
If ∗ is commutative, then G is an Abelian group.
If ∗ is not commutative, then G is called a non-Abelian group.

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Order
Cyclic Groups
Normal Subgroups
Example
(Z,+) is an Abelian group.
(Zn, +n) is an Abelian group.
The set of all nth roots of unity, Un, is an Abelian group
w.r.t. multiplication.
The set of all n × n invertible matrices with real entries,
GL(n, R), called the general linear group, is a non-Abelian
group , w.r.t. matrix multiplication.

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Order
Cyclic Groups
Normal Subgroups
The symmetric group on n letters Sn , n ≥ 3 is a
non- Abelian group.
The group table of S3
* ρ0 ρ1 ρ2 µ1 µ2 µ3
ρ0 ρ0 ρ1 ρ2 µ1 µ2 µ3
ρ1 ρ1 ρ2 ρ0 µ3 µ1 µ2
ρ2 ρ2 ρ0 ρ1 µ2 µ3 µ1
µ1 µ1 µ2 µ3 ρ0 ρ1 ρ2
µ2 µ2 µ3 µ1 ρ2 ρ0 ρ1
µ3 µ3 µ1 µ2 ρ1 ρ2 ρ0

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References
Order
Cyclic Groups
Normal Subgroups
Fact
The equation x2 = e has 4 solutions in Klein-4 group.
The equation x2 = −1 has 6 solutions in Q8.
So a polynomial equation of degree n can have
more than n solutions in a group.

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References
Order
Cyclic Groups
Normal Subgroups
Order
Deﬁnition
The order of an element a in a group is the smallest
positive integer n such that an = e.
The order of a group is the number of elements in the
group.

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Order
Cyclic Groups
Normal Subgroups
Fact
If O(G) is even, then there is atleast one element in G of
order 2.
In a finite group, all elements are of finite order.
There are infinite groups with all elements of finite order.

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Order
Cyclic Groups
Normal Subgroups
Deﬁnition
Let G be an abelian group.
Then Gt = {u ∈ G/O(u) is ﬁnite }, is a subgroup of G,
called the torsion subgroup of G.
If Gt = {0}, then G is said to be torsion free.
If Gt = G, then G is called a torsion group.

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Order
Cyclic Groups
Normal Subgroups
Example
Zn and all ﬁnite Abelian groups are torsion groups.
Z and Z × Z are torsion free groups.
Z × Z2 is neither torsion group nor torsion free group.

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Order
Cyclic Groups
Normal Subgroups
If G is a non-abelian group, there can be two
elements of ﬁnite order whose product is of inﬁnite
order.

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Order
Cyclic Groups
Normal Subgroups
If G is a non-abelian group, there can be two
elements of ﬁnite order whose product is of inﬁnite
order.
Example(i): Consider the following two elements in
GL(2, R).
A =
−1 0
0 1
and B =
−1 −1
0 1
Example(ii): Consider the group G of all permutations on
R, the set of all real numbers and f(x) = −x and
g(x) = 1 − x.

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Order
Cyclic Groups
Normal Subgroups
Theorem (Cauchy)
Let G be a ﬁnite group and let p be a prime number
dividing O(G). Then G has an element of order p and
consequently G has a subgroup of order p.

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Order
Cyclic Groups
Normal Subgroups
Let O(G) = n and d divides n. Does there always exist
an element in G of order d ?

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Order
Cyclic Groups
Normal Subgroups
Let O(G) = n and d divides n. Does there always exist
an element in G of order d ?
Need not!! Look at the following examples.
Klein-4 group.
S3
Z2 × Z4.
Z2 × Z2 × Z2

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Order
Cyclic Groups
Normal Subgroups
Cyclic Groups
Definition
An infinite group is cyclic iff it is isomorphic to (Z,+).
An infinite group is cyclic if and only if it is isomorphic to
all its nontrivial subgroups.
A finite group G is cyclic iff G has an element of order d,
for each d dividing o(G).

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Order
Cyclic Groups
Normal Subgroups
Fact
(Zn, +n) and (Z, +) prototypes of cyclic groups.
Every cyclic group is abelian.
If m is a square free integer, then every Abelian group of
order m is cyclic.
The smallest possible order of a non cyclic group is 4.

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Order
Cyclic Groups
Normal Subgroups
Theorem
The number of subgroups of a cyclic group of order n is the
number of divisors of n.
The number of generators of (Zn, +n) is φ(n), the number
of positive integers less than and relatively prime to n.
A cyclic group with exactly one generator can have atmost
2 elements.
Every group of prime order is cyclic.

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Order
Cyclic Groups
Normal Subgroups
Fact
Every group is the union of its cyclic subgroups.
A group is ﬁnite if and only if it has only ﬁnitely many
subgroups.

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Order
Cyclic Groups
Normal Subgroups
Normal Subgroups
Deﬁnition
A subgroup H is a normal subgroup of G if ghg−1 ∈ H,
∀ g ∈ G and ∀ h ∈ H.
A subgroup H of a group G is normal iﬀ aH = Ha,
∀ a ∈ G.

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References
Order
Cyclic Groups
Normal Subgroups
Example
(2Z, +) is a normal subgroup of (Z, +)
SL(n, R) is a normal subgroup of GL(n, R).
An is a normal subgroup of Sn.

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Order
Cyclic Groups
Normal Subgroups
Theorem
Every subgroup of an abelian group is normal.

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Order
Cyclic Groups
Normal Subgroups
Theorem
Question:Let G be a group such that all its
subgroups are normal. Can we conclude that G is
abelian?

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Order
Cyclic Groups
Normal Subgroups
Theorem
Question:Let G be a group such that all its
subgroups are normal. Can we conclude that G is
abelian?
Answer: NO
Counter example : Q8

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Order
Cyclic Groups
Normal Subgroups
Theorem
Lagrange’s Theorem:
Let G be a ﬁnite group and H be a subgroup of G. Then O(H)
divides O(G).
The converse of Lagrange’s Theorem is true for Abelian
groups.

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Order
Cyclic Groups
Normal Subgroups
Question: Suppose G is a ﬁnite group and it has
subgroups of order d for each d dividing n. Can we
conclude that G is abelian?

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Order
Cyclic Groups
Normal Subgroups
Question: Suppose G is a ﬁnite group and it has
subgroups of order d for each d dividing n. Can we
conclude that G is abelian?
Answer : NO.
Counter examples are
S3
The Quartenion group, Q8

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Order
Cyclic Groups
Normal Subgroups
Theorem
The converse of Lagrange’s Theorem is not generally true.
A4 has no subgroup of order 6.
Proof.
Let H be a subgroup of order 6.
Let a ∈ A4, be of order 3.
Consider the cosets H, aH, a2H. Since o(H) = 6 atleast two of
these cosets must be equal.
H = aH⇒ a∈H
aH = a2H ⇒ a2a−1 ∈ H ⇒ a ∈ H
H = a2H ⇒ a2 ∈ H ⇒ a ∈ H
Thus we get that all elements of order 3,must be in H,which is
a contradiction.(Since there are 8 elements of order 3 in A4).
So A4 cannot have a subgroup of order 6.

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References
Order
Cyclic Groups
Normal Subgroups
Theorem
If H is a normal subgroup of G Then the set
G/H = {aH; a ∈ G} is a group, called factor group, under
the operation (aH)(bH) = abH.

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Order
Cyclic Groups
Normal Subgroups
Deﬁnition
A homomorphism φ from a group G to ¯G is a mapping
from G into ¯G that preserves the group operation.
Ker φ = {x ∈ G/φ(x) = e} .

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Order
Cyclic Groups
Normal Subgroups
Theorem
Ker φ is a normal subgroup of G.
G/Kerφ φ(G).
Every normal subgroup of a group G is the kernel of a
homomorphism of G.

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Order
Cyclic Groups
Normal Subgroups
Example
Z/nZ Zn
R/Z {z ∈ C/|z| = 1}
GL(n, R)/SL(n, R) (R∗, .)
Sn/An U2

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Order
Cyclic Groups
Normal Subgroups
Theorem
Let F be a ﬁnite ﬁeld of q elements. GL(n, q) is the group
of all n × n invertible matrices with entries from F.Then
|GL(n, q)| = (qn − 1)(qn − q)(qn − q2). . .(qn − qn−1).
SL(n, q) is the group of all n × n matrices having
determinant 1,with entries from F.Then
|SL(n, q)| =
(qn − 1)(qn − q)(qn − q2). . .(qn − qn−1)
q − 1
.

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Fundamental Theorem of ﬁnite Abelian g
Direct product of Groups
Consider the group G1 × G2, where G1 and G2 are groups.
If H1, H2 are subgroups of G1 , G2 respectively then
H1 × H2 is a subgroup of G1 × G2.
Is every subgroup of G1 × G2 is of the form H1 × H2?

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Theorem
The group Zm × Zn is isomorphic to Zmn iﬀ m and n are
relatively prime.

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Theorem
Fundamental Theorem of ﬁnite abelian groups:
Every ﬁnite abelian group is a direct product of cyclic groups of
prime-power order. Moreover, the factorisation is unique except
for rearrangement of factors.

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Theorem
If there are r Abelian groups of order m and s Abelian
groups of order n and g.c.d.(m, n) = 1, then there are rs
Abelian groups of order mn.
Let N = pn1
1 pn2
2 ...pnk
k , where pi’s are distinct primes
dividing N. Then the number of abelian groups of order N
is P(n1)P(n2)...P(nk) where P(ni) is the number of
partitions ni.

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How many abelian groups are there of order 4?

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Theorem
For a prime p, every group of order p2 is abelian.
For an odd prime p, every group of order 2p is either Z2p
or the dihedral group Dp.

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Theorem
There are exactly two distinct non-Abelian groups of order
8, the Quartenion group Q8 and the dihedral group D4.

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Order Abelian Groups Non-Abelian Groups
1 Z1
2 Z2
3 Z3
4 Z4, Z2 × Z2
5 Z5
6 Z6 D3
7 Z7
8 Z8, Z4 × Z2, Z2 × Z2 × Z2 D4, Q8
9 Z9, Z3 × Z3
10 Z10 D5

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REFERENCES
[1] Joseph A. Gallian ,”Contemporary Abstract Algebra” ,
Narosa Publishing House.
[2] John.B.Fraleigh, ”A First Course in Abstract Algebra”.
[3] Thomas V.Hungerford, ”Abstract Algebra”

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THANK YOU

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