Correct Answer - Option 3 : (x * y)
2 = x
2 * y
2, for any x, y belonging to G
Option 1: FALSE
If every element in the group is its own inverse, then Group G is abelian group. But converse need not be true.
Option 2: FALSE
If x2 = e (where, x belonging to G and e is the identity element) then Group G is abelian group. But there is no connection between (x = x2) and abelian group.
Option 3: TRUE
A group G is abelian group if and only if (x*y)2 = x2 * y2. (where * is a binary operation)
(x*y)2 = (x*y) (y*x) [G is abelian iff x*y = y*x]
= x*y*y*x = x*y2*x = x*x*y2
= x2 * y2 ( * is binary operation , here it is minus(-) )
So, (G, *) is an abelian group. Then (x * y)2 = x2 * y2, for any x, y belonging to G is true. (True)
Option 4:FALSE
Abelian group can be infinite order. Example - (Z, +)
Note.
x = x-1 for any x belonging to G → G is abelian [only one way]
x2 = e (where, e is the identity element) → G is abelian [only one way]
G is abelian ↔ x*y = y*x
G is abelian ↔ (x*y)2 = x2 * y2