I have to show in a group $G$ of order $4k+2$ the product of the elements cannot be $1$.
What I got so far is that there exists a subgroup $H$ of index $2$ in $G$ (considering left regular representation on $G$).Then $G/H$ is abelian. Now consider the corresponding product of the cosets of $H$, i.e, $x_1x_2\dots x_nH$ where $n=4k+2$, we can arrange this product in such a way that this product becomes $x_1x_2\dots x_mH$ (after some renaming) where $m$ is also odd where each $x_i$ is of order $2$. I cannot proceed further.
Please help me, thank you.