Discrete Computational Structures

DISCRETE COMPUTATIONAL STRUCTURES
MODULE -3

GROUP THEORY

Let G be a non-void set with a binary operation * that assigns to each
ordered pair (a, b) of elements of G an element of G denoted by a * b. We
say that G is a group under the binary operation * if the following three
properties are satisfied:

1) Associativity: The binary operation * is associative i.e. a*(b*c)=(a*b)*c
, ∀ a,b,c ∈ G

2) Identity: There is an element e, called the identity, in G, such that
a*e=e*a=a, ∀ a ∈ G

3) Inverse: For each element a in G, there is an element b in G, called an
inverse of a such that a*b=b*a=e, ∀ a, b ∈ G

Properties of Groups:
The following theorems can understand the elementary features of Groups:
Theorem1:-
1. Statement: - In a Group G, there is only one identity element (uniqueness
of identity) Proof: - let e and e' are two identities in G and let a ∈ G
∴ ae = a ⟶(i)
∴ ae' = a ⟶(ii)
R.H.S of (i) and (ii) are equal ⇒ae =ae'
Thus by the left cancellation law, we obtain e= e'
There is only one identity element in G for any a ∈ G. Hence the theorem is
proved.

2. Statement: - For each element a in a group G, there is a unique element
b in G such that ab= ba=e (uniqueness if inverses)

Proof: - let b and c are both inverses of a a∈ G

Then ab = e and ac = e∵ c = ce {existence of identity element}

⟹ c = c (ab) {∵ ab = e}

, ⟹ c = (c a) b

⟹ c = (ac) b { ∵ ac = ca}

⟹ c = eb

⟹ c = b { ∵ b = eb}Hence inverse of a G is unique.

Theorem 2:-

1. Statement: - In a Group G,(a-1)-1=a,∀ a∈ G

Proof: We have a a-1=a-1 a=eWhere e is the identity element of G

Thus a is inverse of a-1∈ Gi.e., (a-1)-1=a,∀ a∈ G

2. Statement: In a Group G,(a b-1)=b-1 a-1,∀ a,b∈ G

(i) Let ab=ac

Premultiplying a-1 on both sides we get

a-1 (ab)=a-1 (ac)

⟹ (a-1a) b=(a-1 a)c

⟹eb=ec

⟹b=c

Hence Proved.

(ii) Let ba=ca

Post-multiplying a-1 on both sides

⟹(ba) a-1=(ca) a-1

⟹b(aa-1 )=c(aa-1 )

⟹be=ce

⟹b=c, Hence the theorem is proved.