### Group

GROUP

Group ?

A non-empty set G of some elements (a, b, c, etc.), with one or more operations is known as a group.
A set needed to be satisfied following properties to become a group:

1)      Closure Property-
a.b ∈  G , ∀ a, b ∈  G

2)      Associative Property-
(a . b) . c = a . (b . c), ∀ a, b, c ∈  G

3)      Existence of Identity-
e →  identity element
e.a = a = a.e, ∀  a ∈ G

4)      Existence of Inverse-
a-1→  inverse of a
a.a-1 = e = a-1.a , ∀  a ∈ G

ABELIAN OR COMMUTATIVE GROUP

A set needed to be satisfied following properties to become an abelian group:

1)      Closure Property-
a.b ∈  G , ∀ a, b ∈  G

2)      Associative Property-
(a . b) . c = a . (b . c), ∀ a, b, c ∈  G

3)      Existence of Identity-
e →  identity element
e.a = a = a.e, ∀  a ∈ G

4)      Existence of Inverse-
a-1 →   inverse of a
a.a-1 = e = a-1.a , ∀  a ∈ G

5)      Commutativity-
a.b = b.a , ∀  a , b ∈ G

SUBGROUP

A subgroup is a subset H of group elements of a group G that satisfies all the four properties of a group.

“ H is a subgroup of G” can be written as H ⊆ G

A subgroup H of a group G, where H ≠ G, is known as proper subgroup of G.

Problems based on above study:

Prob. Show that the set I of all integers (…, -4, -3, -2, -1, 0, 1, 2, 3, 4,…}
Is a group with respect to the operation of addition of integers?
Solution:

Prob. Let ({a, b}, * ) be a semigroup where a*a = b show that-
i.            a*b = b*a
Solution:

Prob. Let (A, *) be a semigroup. Show that for a, b, c ∈  A, if a*c = c*a and b*c = c*b, then (a*b)*c = c*(a*b).
Solution:

Prob. Suppose (A, *) be a group, show that (A, *) is an abelian group and only if a3 * b3 = (a*b)3 for all a and b in A.
Solution:

Prob. Prove that the set Z of all integers with binary operations defined by-
a*b = a+b+1 , ∀  a, b ∈  z.  Is an abelian group.

Prob. Prove that the cube roots of unity namely ( 1, w, w2) abelian under multiplication of complex numbers.
Solution:

Prob. Prove that the set G = {0, 1, 2, 3, 4, 5} is a finite abelian group of order 6 with respect to addition modulo 6.
Solution:

Prob. Prove that the set G = {1, 2, 3, 4, 5, 6} is a finite abelian group of order 6 with respect to multiplication modulo 7.
Solution:

Prob. Which of the following property/ies a Group G must hold, in order to be an Abelian group? (a) The distributive property (b) The commutative property (c) The symmetric property. [CBSE NET December 2015]