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Let n > 1 be a fixed integer and let G be a group. If the set H = {x in G : |x| = n} together with the identity forms a subgroup of G, what can be said about n?

I know that n must be prime, but I can't figure out why that would be. The elements of h only have order 1 or n and no others and the order of an element divides the order of the (sub)group. Are we making the assumption that since there are no other divisors of the group order, n must be prime? Or is it something else...

Any insight would be appreciated.

Cheers,

W. =)

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# Homework Help: Subgroups and prime order elements

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