Question

In: Advanced Math

Let LaTeX: GG be an abelian group. Let LaTeX: H = { g \in G \mid...

Let LaTeX: GG be an abelian group. Let LaTeX: H = { g \in G \mid g^3 = e }H = { g ∈ G ∣ g 3 = e }. Prove or disprove: LaTeX: H \leq GH ≤ G.

Expert Solution

Given that is an abelian group.

Then .

Also given that , being the identity element.

Theorem: a subset of a group is a subgroup of if and only if .

To prove :

Let

Then

Now

Since are also in is a subset of

Then .

Therefore we get

This is true for all .

Hence by using the above theorem we can conclude that

is a subgroup of , i.e, .

Colby Messinger answered 1 year ago

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