- Thread starter
- Moderator
- #1

- Jan 26, 2012

- 995

-----

**Problem**: Let $G$ be a finite group and for some fixed $n\in\mathbb{N}$ let $H=\{g\in G: g^n=1\}$. If $G$ is abelian, prove that $H$ is a characteristic subgroup of $G$ (Recall that $H$ is a characteristic subgroup of $G$ if (i) $H\leq G$ and (ii) for every $f\in\text{Aut}(G)$, $f(H)=H$). If $G$ isn't abelian, find a specific $G$ and $n\in\mathbb{N}$ that shows $H$ is not a characteristic subgroup in $G$ (in this case, it just suffices to show that $H$ is not a subgroup of $G$).

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!