Let G be an Abelian group. Prove or disprove that

H={g^2|g is a member of G}

is a subgroup of G.

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- Oct 25th 2008, 07:16 AMmandy123prove or disprove
Let G be an Abelian group. Prove or disprove that

H={g^2|g is a member of G}

is a subgroup of G. - Oct 25th 2008, 07:41 AMPaulRS
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'