(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let G be an abelian group, k a fixed positive integer, and H = {a is an element of G; |a| divides k}. Prove that H is a subgroup of G.

2. Relevant equations

Definition of groups, subgroups, and general knowledge of division algorithm.

3. The attempt at a solution

I know that to prove H is a subgroup of G, I need to:

1) prove that it is closed (so if a and b are in H, ab is in H)

2) prove that if a is in H, then a^{-1}is in H as well.

I'm somewhat confused on how to proceed. To prove the first part, I suppose that ab is an element of H. Then we can write abn = k for some integer n. Because G is abelian, we can use associativity and commutativity to write (a)(bn) = k and (b)(an) = k. This clearly shows that a divides k and b divides k.

The problem is that I'm not sure if I went the wrong direction - meaning I supposed ab was in H, and proved a and b are in H.

I'm also at a loss on how to approach the second part. It's driving me mad because this is an "easy" problem and it's the one that's stumping me on this problem set.

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# Proving H is a subgroup of G, given

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