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

GROUPS
References
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

GROUPS
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.

GROUPS
References
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.

GROUPS
References
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

GROUPS
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.

GROUPS
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.

GROUPS
References
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.

GROUPS
References
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.

GROUPS
References
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.

GROUPS
References
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.

GROUPS
References
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.

GROUPS
References
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.

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

GROUPS
References
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

GROUPS
References
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).

GROUPS
References
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.

GROUPS
References
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.

GROUPS
References
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.

GROUPS
References
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.

GROUPS
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.

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

GROUPS
References
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?

GROUPS
References
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

GROUPS
References
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.

GROUPS
References
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?

GROUPS
References
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

GROUPS
References
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.

GROUPS
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.

GROUPS
References
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} .

GROUPS
References
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.

GROUPS
References
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

GROUPS
References
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
.

GROUPS
References
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?

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

GROUPS
References
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.

GROUPS
References
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.

GROUPS
References
How many abelian groups are there of order 4?

GROUPS
References
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.

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

GROUPS
References
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

GROUPS
References
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”

GROUPS
References
THANK YOU

Group Theory

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Group Theory

Similar to Group Theory (20)

Recently uploaded

Recently uploaded (20)

Group Theory