A group is an algebraic structure (G, *) in which the binary operation * on G satisfies the following conditions:

**Condition-1**: For all a, b, c, ∈ G

a* (b * c) = (a * b) * c (associativity)

**Condition-2: **There exists an elements e ∈G such that for any a ∈G

a* e= e * a = a (existence of identity)

**Condition-3:** For every a ∈G, there exists an element denoted by in G such that

is called the inverse of a in G.

is called the inverse of a in G.

**Example: (Z, +) is a group where Z denote the set of integers.**

**Example: (R, +) is a group where R denote the set of real numbers.**

## Different types of groups

**Abelian group**

Let (G, *) be a group. If * is commutative that is

a* b = b * a for all a, b ∈G then (G, *) is called an Abelian.

**Finite group-**

G is said to be a finite group, if the set G is a finite set.

**Infinite group-**

A group G, which is not finite is called an infinite group.

**Order of a finite group**

The order of a finite group (G, *) is the number of distinct element in G. The order of

G is denoted by O (G) or by |G|.

**Example**

**If G = {1, -1, i, -i} where , then show that G is an abelian group with respect to multiplication as a binary operation.**

Sol.

First we will construct a composition table-

. | 1 | -1 | I | -i |

1 | 1 | -1 | I | -i |

-1 | -1 | 1 | -i | i |

i | i | -i | -1 | 1 |

-i | -i | I | 1 | -1 |

It is clear from the above table that algebraic structure (G, .) is closed and satisfies the following conditions.

Associativity- For any three elements a, b, c ∈G (a ⋅b) ⋅c = a ⋅(b ⋅c)

Since

1 ⋅(−1 ⋅i) = 1 ⋅−i= −i

(1 ⋅−1) ⋅i= −1 ⋅i= −i

⇒1 ⋅(−1 ⋅i) = (1 ⋅−1) i

Similarly, with any other three elements of G the properties hold.

∴ Associative law holds in (G, ⋅)

Existence of identity: 1 is the identity element (G, ⋅) such that 1 ⋅a = a = a ⋅1 ∀a ∈G

Existence of inverse: 1 ⋅1 = 1 = 1 ⋅1 ⇒1 is inverse of 1

(−1) ⋅(−1) = 1 = (−1) ⋅(−1) ⇒–1 is the inverse of (–1)

i⋅(−i) = 1 = −i⋅i⇒–iis the inverse of iin G.

−i⋅i= 1 = i⋅(−i) ⇒iis the inverse of –iin G.

Hence the inverse of every element in G exists.

Thus all the axioms of a group are satisfied.

Commutativity: a ⋅b = b ⋅a ∀a, b ∈G hold in G

1 ⋅1 = 1 = 1 ⋅1, −1 ⋅1 = −1 = 1 ⋅−1

i⋅1 = i= 1 ⋅i; i⋅−i= −i⋅i= 1 = 1 etc.

Commutative law is satisfied.

Hence (G, ⋅) is an abelian group.

**Example**

**Prove that the set Z of all integers with binary operation * defined by a * b = a + b + 1 ****∀****a, b ****∈****G is an abelian group.**

**Sol: **Sum of two integers is again an integer; therefore a +b ∈Z ∀a, b ∈Z

⇒a +b + 1 ⋅∈Z ∀a, b ∈Z

⇒Z is called with respect to *

Associative law for all a, b, a, b ∈G we have (a * b) * c = a * (b * c) as

(a* b) * c = (a + b + 1) * c

= a + b + 1 + c + 1

= a + b + c + 2

Also

a* (b * c) = a * (b + c + 1)

= a + b + c + 1 + 1

= a + b + c + 2

Hence (a * b) * c = a * (b * c) ∈a, b ∈Z.

**Subgroup**

Let (G, *) be a group and H, be a non-empty subset of G. If (H, *) is itself is a group, then (H, *) is called sub-group of (G, *).

**Example-Let a = {1, –1, i, –i} and H = {1, –1}**

**G and H are groups with respect to the binary operation, multiplication.**

**H is a subset of G, therefore (H, X) is a sub-group (G, X).**

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