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"Let G be a group and let G_{n}={g^{n}|g ε G}. Under what hypothesis about G can we show that G_{n}is a subgroup of G?

The set G_{n}is taking each element of G and raising it to a fixed number. I started my investigation by examining what happens if I take n=3 and considering the groupsZ_{4}and the Klein 4-group. I noticed that G_{n}did not create a subgroup with the integers modulo 4, but it did create one with the Klein 4-group. Thus, I believe that in order for G_{n}be a subgroup, we need the condition that G must NOT be cyclic.

That being said, I have a proof, which seems to work, but nowhere did I use the fact that G is not cyclic. I have an outline below.

Thrm: If G is a group and is not cyclic, then G_{n}={g^{n}|g ε G} is a subgroup.

Proof (SKETCH): Let G be a group and suppose G is not cyclic. We wish to show G_{n}={g^{n}|g ε G} is a subgroup.

*Associativity*

... inherited from G

*Identity*

... The identity element e from G is in G_{n}, and e raised to some fixed number n is still e.

*Inverse*

... a and a^{-1}are in G... a^{n}ε G_{n}.. a^{-1}^{n}= a^{-n}ε G_{n}... a^{n}a^{-n}=a^{-n}a^{n}=e

The above is just an outline. But, even with all the details, I never use the fact that G is not cyclic. I must be doing something wrong. Please help! Thanks

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# Homework Help: Problem concerning cyclic groups.

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