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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, 781.png a, b, c  G
  2. e  G  a * e = e * a = a, 786.pnga  G
  3. a  G
    b  G  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  G , then 791.png

Theorem 4 :

If (G, .) be a group and a, b  G, then (ab)–1 = b–1a–1

Theorem 5 :

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


Theorem 6 :

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

I. a, b, c  G, ab = ac

b = c

II. a, b, c,  G, ba = ca

b = c.

Definition 4.

If (G, .) be a gorup and a  G, n  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 = (a–n)–1

Theorem 7.

Let (G, .) be a group and a  G.

If m, n  Z, then

  1. am . an = am+n = anam
  2. 802.png 

Definition 5.

Let (G, .) be a group. An element a  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 .

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!

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