Group Theory

GROUP
BY
Durgesh Chahar (M.Phil Scholar)
I.B.S Khandari agra
1

History
The term group was coined by Galois
around 1830 to described sets functions
on finite sets that could be grouped
together to form a closed set. The
modern definition of the group given by
both Heinrich Weber and Walter Von Dyck
in 1882, it did not gain universal
acceptance until the twentieth century.
2

Binary Operation :-
Let 𝐺 be a set. A binary operation on 𝐺 is a
function that assigns each order pair of
elements of 𝐺 an element of 𝐺.
𝑓 ∶ 𝐺 × 𝐺 → 𝐺
 Remark :
o is a binary operation on 𝐺 iff aοb Є 𝐺. 3

Algebraic Structure :-
 A non empty set together with one or more
than one binary operation is called
algebraic structure.
Examples :-
1. (R,+, ∙) is an algebraic structure.
2. (N, +) , (Z, +), (Q, +) are algebraic
structures.
4

Group :-
A non empty set 𝐺 together with an operation o is called a group if the following
conditions are satisfied :
• Closure axiom,
∀ 𝑎, 𝑏 ∈ 𝐺 ⇒ 𝑎𝑜𝑏 ∈ 𝐺.
• Associative axiom,
𝑎𝑜𝑏 𝑜𝑐 = 𝑎𝑜(𝑏𝑜𝑐) ∀ 𝑎, 𝑏, 𝑐 ∈ 𝐺
• Existence of identity,
∃ an element 𝑒 ∈ 𝐺, called identity
𝑎𝑜𝑒 = 𝑒𝑜𝑎 = 𝑎 ∀ 𝑎 ∈ 𝐺.
• Existence of inverse,
𝑎 ∈ 𝐺 , ∃ 𝑎−1
∈ 𝐺 𝑠. 𝑡
𝑎−1 𝑜𝑎 = 𝑎𝑜𝑎−1 = 𝑒
This 𝑎−1 is called inverse of 𝑎. 5

Abelian Group :-
A group 𝐺, 𝑜 is called abelian group or commutative group if
𝑎𝑜𝑏 = 𝑏𝑜𝑎 ∀ 𝑎, 𝑏 ∈ 𝐺.
Examples :-
1. ℤ, + , ( ℚ , +) , (ℝ, +) all are commutative group.
2. ℚ0,∙ , ℝ0,∙ are commutative group. 1 is an identity ,1
𝑎
is the inverse of 𝑎 in each case.
3. The set of all 𝑚 × 𝑛 matrics (real and complex) with
matrix addition as a binary operation is commutative
group. The zero matric is the identity element and the
inverse of matric of A is –A. 6

Quaternion Group :-
𝐺 = { ±1, ±𝑖, ±𝑗, ±𝑘 } define a binary operation of multiplication as
𝑖2
= 𝑗2
= 𝑘2
= −1 , 𝑖𝑗 = −𝑗𝑘 = 𝑘, 𝑘𝑖 = −𝑖𝑘 = 𝑗, 𝑗𝑘 = −𝑘𝑗 = 𝑖 .
The red arrows represent multiplication on the
right by 𝑖, and the green arrows represent
multiplication on the right by 𝑗.
This is non abelian group for this operation.
This is called Quaternion group.
7

Klein’s four group
 Let 𝐺 = ( 𝑒, 𝑎, 𝑏, 𝑐 ) with operation 𝑜 defined by the
following table :
𝒐 e a b c
e e a b c
a a e c b
b b c e a
c c b a e
8

Theorem :-
Uniqueness of identity
 The identity 𝑒 in a group always unique.
Proof-
If possible, suppose that 𝑒 and 𝑒′ are two identity elements in a
group 𝐺.
𝑒 is an identity element
⇒ 𝑒𝑒′
= 𝑒′
𝑒 = 𝑒
′
𝑎𝑒 = 𝑒𝑎 = 𝑎
𝑒′
is an identity element
⇒ 𝑒𝑒′ = 𝑒′ 𝑒 = 𝑒 [ 𝑎𝑒′ = 𝑒′ 𝑎 = 𝑎]
these statements prove that 𝑒 = 𝑒𝑒′
= 𝑒′
𝑒 = 𝑒′
from which, we get 𝑒 = 𝑒′.
9

Theorem :-
The cancellation laws
 Suppose, 𝑎, 𝑏, 𝑐 are arbitrary elements of a group 𝐺. Then
1. 𝑎𝑏 = 𝑎𝑐 ⇒ 𝑏 = 𝑐 ( left cancellation )
2. 𝑏𝑎 = 𝑐𝑎 ⇒ 𝑏 = 𝑐 (right cancellation )
Proof :-
Let 𝑒 be the identity element in a group 𝐺. Let 𝑎, 𝑏, 𝑐 ∈ 𝐺 be arbitrary
𝑎𝑏 = 𝑎𝑐
⇒ 𝑎−1
𝑎𝑏 = 𝑎−1
(𝑎𝑐)
⇒ 𝑎−1 𝑎 𝑏 = 𝑎−1 𝑎 𝑐 [by associative law]
⇒ 𝑒𝑏 = 𝑒𝑐
⇒ 𝑏 = 𝑐
10

Again
𝑏𝑎 = 𝑐𝑎
⇒ 𝑏𝑎 𝑎−1
= 𝑐𝑎 𝑎−1
⇒ 𝑏 𝑎𝑎−1 = 𝑐 𝑎𝑎−1
⇒ 𝑏𝑒 = 𝑐𝑒
⇒ 𝑏 = 𝑐
Example :-
1. The positive integer form a cancellative semigroup under addition.
2. The non-negative integers form a cancellative monoid under addition.
3. The cross product of two vectors does not obey the cancellation law.
if 𝑎 × 𝑏 = 𝑎 × 𝑐, then it does not follow that 𝑏 = 𝑐 even if 𝑎 ≠ 0.
11

4. Matrix multiplication also does not necessary obey the cancellation law.
AB = BC and A ≠ 0
Consider the set of all 2 × 2 matrices with integer coefficients. The matrix
multiplication is defined by
𝑎 𝑏
𝑐 𝑑
𝑎′ 𝑏′
𝑐′
𝑑′ =
𝑎𝑎′
+ 𝑏𝑐′ 𝑎𝑏′
+ 𝑏𝑑′
𝑐𝑎′
+ 𝑑𝑐′ 𝑐𝑏′
+ 𝑑𝑑′
It is associative, and
1 0
0 1
is identity but the cancellation law does not follow
0 1
0 0
0 2
0 0
=
0 0
0 0
and
0 1
0 0
3 0
0 0
=
0 0
0 0
This implies
0 1
0 0
0 2
0 0
=
0 1
0 0
3 0
0 0
but
0 2
0 0
≠
3 0
0 0
12

Theorem :-
Uniqueness of inverse
 The inverse of each element of a group is unique.
Proof :-
If possible, let 𝑎 and 𝑏 be two elements of a group 𝐺, so that
𝑏𝑎 = 𝑎𝑏 = 𝑒 ...(1)
𝑐𝑎 = 𝑎𝑐 = 𝑒 ...(2)
𝑒 be an identity in 𝐺.
𝑏𝑎 = 𝑒 = 𝑐𝑎
or 𝑏𝑎 = 𝑐𝑎
𝑏 = 𝑐 [by right cancellation law.]
13

Theorem :-
 The left identity is also the right identity.
Proof:-
Let e be the left identity of a group 𝐺 and let 𝑎 ∈ 𝐺 be arbitrary. Then
𝑒𝑎 = 𝑎 …………… (1)
To prove that 𝑒 is also that right identity. It suffices to show that
𝑎𝑒 = 𝑎
suppose 𝑎−1
is the left inverse of a, then
𝑎−1
is the left inverse of a, then
𝑎−1
𝑎 = 𝑒 …………… (2)
by associative law in 𝐺.
𝑎−1
𝑎𝑒 = 𝑎−1
𝑎 𝑒 = 𝑒 [ by (2)]
= 𝑒 = 𝑎−1
𝑎 [ again by (2)]
𝑎−1
𝑎𝑒 = 𝑎−1
𝑎 ⇒ 𝑎𝑒 = 𝑎 [ by left cancellation law]
14

Theorem :-
Reverse rule
 Let 𝑎 and 𝑏 be the elements of a group 𝐺. Then
then (𝑎𝑏)−1
= 𝑏−1
𝑎−1
.
Proof:-
consider arbitrary elements 𝑎 and 𝑏 of a group 𝐺.
Since 𝑏−1
and 𝑎−1
are inverse of 𝑎 and 𝑏 respectively.
𝑏−1
𝑏 = 𝑏𝑏−1
= 𝑒, and 𝑎−1
𝑎 = 𝑎𝑎−1
= 𝑒 ………….(1)
Hence, by associativity law,
(𝑎𝑏)(𝑏−1
𝑎−1
) = 𝑎 𝑏𝑏−1
𝑎−1
= 𝑎 𝑒𝑎−1
= 𝑎𝑎−1
= 𝑒 ……………(2)
This ⇒ (𝑎𝑏)−1
= 𝑏−1
𝑎−1
.
Generalizing this result, we obtain
(𝑎𝑏𝑐 … )−1
= … 𝑐−1
𝑏−1
𝑎−1
15

Theorem:-
 If let 𝐺 be a group and 𝑎 ∈ 𝐺 then (𝑎−1)−1 = 𝑎.
Proof:-
let 𝑎−1
be the inverse of an element 𝑎 of a group 𝐺, then
𝑎−1 𝑎 = 𝑒 ……………(1)
to prove that the inverse of 𝑎−1 is 𝑎,
premultiplying (1) by (𝑎−1
)−1
,
[(𝑎−1)−1 𝑎−1] 𝑎 = (𝑎−1)−1 𝑒, by associative law
𝑒𝑎 = (𝑎−1)−1
𝑎 = (𝑎−1)−1
Remark - 𝑒−1 = 𝑒.
16

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