If $G$ is a finite group with the property that every non-identity element has even order, what could be said about $G$? For example the most trivial property might be that $G$ is a group of even order. I was wondering if more could be said about the structure of $G$.
Thanks!
The condition holds if and only if $G$ is a $2$-group; that is, if the order of $G$ is a power of $2$.
To see this, note that Cauchy's theorem implies that $|G|$ cannot be divisible by an odd prime.