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 x^{2} = 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 = x^{2}) and abelian group.

__Option 3: __TRUE

A group G is abelian group if and only if (x*y)^{2} = x^{2} * y^{2}. (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*y^{2}*x = x*x*y^{2}

= x^{2} * y^{2 } ( * is binary operation , here it is minus(-) )

So, (G, *) is an abelian group. Then (x * y)^{2} = x^{2} * y^{2}, 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]

x^{2 }= 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} = x^{2} * y^{2}