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.
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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