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.
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Matt Samuel
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Billie
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2Hint: What will the order of $G/\langle a\rangle$ be? – Tobias Kildetoft May 21 '15 at 19:45
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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?
Matt Samuel
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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
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@TobiasKildetoft true, but you still need Lagrange's theorem with that argument. – Matt Samuel May 21 '15 at 19:49
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@TobiasKildetoft Also your hint is a standard exercise, and that's the way I would have solved it. I agree this hints is very nice, and too bad only one wrote an answer as I would have upvoted both. – Timbuc May 21 '15 at 19:49
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@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
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