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Let G be an abelian group, let H = {x in G | (x^3) = eg},...

Let G be an abelian group, let H = {x in G | (x^3) = eg}, where eg is the identity of G. Prove that H is a subgroup of G.

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Answer #1

A subgroup is a subset of the group, which is a group onto itself under the same operation. To check whether a given subset is a subgroup we check using one step subgroup test, two step subgroup test or the finite subgroup test.

In this question we know,

where .

We know, . So . And .

Now, let .

This implies, .

Consider .

Since, the group is abelian, this is equal to

((Since, by definition of H)).

So, since , we get .

Therefore, H is a subgroup by one step subgroup test.

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