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Let $G$ be a group of order $2n$, where $n$ is odd; show that there exists a normal subgroup $H$ of index $2$ and show that this subgroup is characteristic as well.

My work:

I was able to notice that there exist an element $x$ of period $2$ in $G$ (by Sylow theorems) and considering the usual injective homomorphism induced by left multiplication: $$\varphi:G\to \text{Sym}(G)$$ we can see that the period of $\varphi (x) =2$ which implies that $\varphi (x)$ is odd so $G$ does not inject into $A_{2n}$ but I don't know how to continue.

Thanks in advance

terran
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iki
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  • Yes in part, but that doesn't show that the normal subgroup is char as well as I don't really follow how showing that there exists an odd permutation is sufficient for showing the existence of a normal subgroup of order n (order m in the question) – iki Aug 29 '23 at 09:10

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