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 
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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