Groups

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

Note: If a group has the property that a*b=b*a i.e., commutative law holds then the group is called an abelian.

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=e

Where e is the identity element of G

Thus a is inverse of a^-1∈ G

i.e., (a^-1)^-1=a,∀ a∈ G

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

Proof: - By associatively we have

(b^-1 a^-1)ab=b^-1 (a^-1 a)b
⟹(b^-1 a^-1)ab=b^-1 (e)b         {∵a^-1 a=e}
⟹(b^-1 a^-1)ab=b^-1 b         {∵eb=b}
⟹(b^-1 a^-1)ab=e,         {∵b^-1 b=e}

Similarly

(ab) (b^-1 a^-1)=a(b b^-1) a^-1
⟹(ab) (b^-1 a^-1)=a (e) a^-1
⟹(ab) (b^-1 a^-1)=a a^-1
⟹(ab) (b^-1 a^-1)=e         {∵aa^-1=e}
Thus ( b^-1 a^-1)ab=(ab)(b^-1 a^-1)=e
∴ b^-1 a^-1 is the inverse of ab
i.e., b^-1 a^-1= a b^-1

Hence the theorem is proved.

Theorem3:-

In a group G, the left and right cancellation laws hold i.e.

(i) ab = ac implies         b=c

(ii) ba=ca implies         b=c

Proof

(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.

Finite and Infinite Group:

A group (G, *) is called a finite group if G is a finite set.

A group (G, *) is called a infinite group if G is an infinite set.

Example1: The group (I, +) is an infinite group as the set I of integers is an infinite set.

Example2: The group G = {1, 2, 3, 4, 5, 6, 7} under multiplication modulo 8 is a finite group as the set G is a finite set.

Order of Group:

The order of the group G is the number of elements in the group G. It is denoted by |G|. A group of order 1 has only the identity element, i.e., ({e} *).

A group of order 2 has two elements, i.e., one identity element and one some other element.

Example1: Let ({e, x}, *) be a group of order 2. The table of operation is shown in fig:

*	e	x
e	e	x
x	x	e

The group of order 3 has three elements i.e., one identity element and two other elements.