# (G, *) is an abelian group. then

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(G, *) is an abelian group. then
1. x = x-1 for any x belonging to G
2. x = x2 for any x belonging to G
3. (x * y)2 = x2 * y2, for any x, y belonging to G
4. G is of finite order

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Correct Answer - Option 3 : (x * y)2 = x2 * y2, 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