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Let $G$ be a group and $$N:=\langle\{g^2:g\in G\}\rangle$$ Prove that $N$ is a normal subgroup.

I am quite not sure how to start on this exercise. I heard that this can be proven by using the notion of an index of a group but since we did not cover this in the lecture I think it should not be used. Has someone a hint for a more basic solution?

TheGeekGreek
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1 Answers1

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$x g^2 x^{-1} = (x g x^{-1}) (x g x^{-1}) = (x g x^{-1})^{2} \in N$.

user8960
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