Prove that $H$ is a abelian subgroup of odd order

Question is:

Let $G$ be a group of order 2n. Suppose half of the element of G are of order 2 and the other half form a subgroup $H$ of order n . Prove that $H$ is of odd order and is an abelian subgroup of $G$

What could i see is..

if we prove that order of every element of $H$ is odd , then order of $H$ is odd . Also i am unable to use the fact that half of the element of $G$ are of order 2.

Please help me to clear this.

Thank You.

Solution 1:

I assume no element of $H$ has order $2$. Then every nonidentity element of $H$ can be paired with its own inverse. Since $e\in H$ as well, you get that $H$ is a subgroup of odd order.

Moreover, $[G:H]=2$, so $H$ is normal in $G$. Pick $g\in G\setminus H$. This induces a conjugation automorphism of $H$, $$ \varphi_g\colon H\to H:h\mapsto ghg^{-1}=ghg $$ since $g$ has order $2$. In particular, for any $h\in H$, $$ h\varphi_g(h)=hghg=(hg)^2=e $$ since $hg\notin H$, lest $g\in H$. So you conclude $\varphi_g(h)=h^{-1}$. It follows that $H$ is abelian, for if $h,k\in H$, $$ k^{-1}h^{-1}=(hk)^{-1}=\varphi_g(hk)=\varphi_g(h)\varphi_g(k)=h^{-1}k^{-1}. $$