# Simple group with Klein four Sylow

If $G$ is a simple group, with a Sylow $2$-subgroup isomorphic to the Klein four group $\mathbb{Z}_2 \times \mathbb{Z}_2$, then I want to show that any two involutions in a given Sylow $2$-subgroup are conjugate in $G$.

Any help would be appreciated.

Note: A solution/method in relatively elementary terms would be appreciated — I am alright with some machinery as long as you explain it.

• I think you need Burnside's Transfer Theorem to prove that. Do you know it? – Derek Holt Oct 30 '14 at 8:28
• @DerekHolt I'm afraid not. If you could explain it I would be happy to understand it as part of the solution. – user151882 Oct 30 '14 at 13:29