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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.

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  • $\begingroup$ I think you need Burnside's Transfer Theorem to prove that. Do you know it? $\endgroup$ – Derek Holt Oct 30 '14 at 8:28
  • $\begingroup$ @DerekHolt I'm afraid not. If you could explain it I would be happy to understand it as part of the solution. $\endgroup$ – user151882 Oct 30 '14 at 13:29
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I think you gate this from Sylow's 2nd theorem

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    $\begingroup$ That tells you that the Sylow Subgroups are conjugate. This question is dealing with the elements inside a single Sylow Subgroup. $\endgroup$ – Mark Bennet Oct 30 '14 at 7:41

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