I was reading the paper "Central elements in core-free groups" of Glauberman: http://www.sciencedirect.com/science/article/pii/0021869366900305
He defines the core of a finite group $G$ to be the largest normal subgroup of $G$ of odd order.
A weaker version of theorem 1 in the paper states the following: let $G$ be a group of even order with trivial center and trivial core, and $t \in G$ of order 2. Then there exists a $g \in G$ such that the commutator $[t, g]$ has even order.
This being weaker than theorem 1, I was wondering if there is another proof, which doesn't use as much character theory, or at least which is somewhat shorter. By lemma 2, an equivalent statement would be to say that with $t$ and $G$ as above, if $S$ is a 2-Sylow subgroup of $G$ containing $t$ then $S$ contains another $G$-conjugate of $t$.