# Let $G$ a finite group with order of $2p$, where $p > 2$ is prime. given that there's $a \in Z(G)$ such that $o(a) = 2$. Prove: $G$ is abelian. [duplicate]

Homework question:

Let $G$ a finite group with order of $2p$, where $p > 2$ is prime. given that there's $a \in Z(G)$ such that $o(a) = 2$. Prove: $G$ is abelian.

Can you give me some hints and help to solve it? Thanks in advance.

## marked as duplicate by TravisJ, graydad, colormegone, Rebecca J. Stones, Jonas MeyerMay 22 '15 at 3:30

• Hint: What will the order of $G/\langle a\rangle$ be? – Tobias Kildetoft May 21 '15 at 19:45
Hint: by Cauchy's theorem there is an element $x$ of order $p$. What does the subgroup generated by $a$ and $x$ look like, and what is its order?
• Ahh, even simpler than my version. Though Cauchy is not needed, just that there is an element not of order $2$ (and if they are all of order $2$, then the group being abelian is a standard exercise). – Tobias Kildetoft May 21 '15 at 19:47
• @TobiasKildetoft so the idea is to make the G = { $x^i$, $a*x^i$ } for $0 \le i<p$ ? – Billie May 21 '15 at 21:45