Let G be an Abelian group. Prove or disprove that
H={g^2|g is a member of G}
is a subgroup of G.
Indeed it's true, take then (*)
Where we've used the fact that G is abelian and that the associativity holds.
So if and then by (*) we have
Also the identity belongs to H since
And each element has an inverse, say we have then for some
Now has an inverse, namely
So note that thus is the inverse of a'