# 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

- a * (b * c) = (a * b) * c, a, b, c âˆˆ G
- âˆƒe âˆˆ G â‡’ a * e = e * a = a, a âˆˆ G
- 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} = b^{â€“1}a^{â€“1}

Theorem 5 :

If (G, .) be a group and a_{1}, a_{2}, .... a_{n} 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 a^{n} is defined as follows :

- a
^{0}=e - If n > 0, then a
^{1}= a; a^{n+1}= a^{n}. a - If n < 0, then a
^{n}= (a^{â€“n})^{â€“1}

**Theorem 7.**

Let (G, .) be a group and a âˆˆ G.

If m, n âˆˆ Z, then

- a
^{m}. a^{n}= a^{m+n}= a^{n}a^{m}

**Definition 5.**

Let (G, .) be a group. An element a âˆˆ G is called idempotent if a^{2} = 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 P_{n} or S_{n}.

**
Note :** 0(S

_{n}) = n!