# Group

A semigroup with identity and in which every element is invertible is called a group.

Definition 1

Let G be a non-empty set and * be a binary operation on G. Then algebraic system (G, *) is called a group if

1. a * (b * c) = (a * b) * c,  a, b, c âˆˆ G
2. âˆƒe âˆˆ G â‡’ a * e = e * a = a, a âˆˆ G
3. a âˆˆ G
â‡’ âˆƒb âˆˆ G â‡’ a * b = b * a = e
• The element e in condition II of the definition of a group is called identity element of the group.
• The element b corresponding to a in condition III of the definition of a group is called inverse of a in the group.

Definition 3 :

A group (G, *) is said to be a non-abelian group, if (G, *) is not abelian.

# Properties Of A Group

Theorem 1 :

In a group, identity element is unique.

Theorem 2 :

In a group, inverse of every element is unique.

Theorem 3 :

If (G, .) be a group and a âˆˆ G , then

Theorem 4 :

If (G, .) be a group and a, b âˆˆ G, then (ab)â€“1 = bâ€“1aâ€“1

Theorem 5 :

If (G, .) be a group and a1, a2, .... an âˆˆ G, then

Theorem 6 :

Cancellation laws hold in a group, i.e. If (G, .) is a group then

I. a, b, c âˆˆ G, ab = ac

â‡’ b = c

II. a, b, c, âˆˆ G, ba = ca

â‡’ b = c.

Definition 4.

If (G, .) be a gorup and a âˆˆ G, n âˆˆ z, then an is defined as follows :

1. a0 =e
2. If n > 0, then a1 = a; an+1 = an . a
3. If n < 0, then an = (aâ€“n)â€“1

Theorem 7.

Let (G, .) be a group and a âˆˆ G.

If m, n âˆˆ Z, then

1. am . an = am+n = anam
2.

Definition 5.

Let (G, .) be a group. An element a âˆˆ G is called idempotent if a2 = a.

Definition 6.

A group (G, .) is called finite group if G is a finite set The number of different elements in G is callled order of the fininte group (G, .). It is denoted by O(G).

Definition 7.

A group (G, .) is called infinite group if G is an infinite set. The order of an infinite group is defined to be âˆž.

Definition 8.

If S is a finite set containing n elements, then group of all bijections on S is called a permutation group or symmetric group. It is denoted by Pn or Sn.

Note :
0(Sn) = n!