In the paper, Janko shows the simplicity of Janko group $J_1$ at the Lemma 2.1. In this proof it says "By a transfer theorem all involutions are conjugate in $G$", but I cannot understand.
- Some propositions are named "transfer theorem", such as Burnside's transfer theorem, Thompson's transfer theorem, etc... . I want to know which proposition is used.
- I found an Additional description for the proof, but this part of proof is not correct (because the author uses the result of the Janko's paper, and the original proof doesn't use any specific calculations). I want to know theoretic approach of the problem.
The group in question is simple and has elementary abelian Sylow 2-subgroups of order 8.
A crucial fact that you need to know is that, if finite group $G$ has an abelian Sylow $p$-subgroup $P$, then $p$-transfer is controlled by $N_G(P)$ - see here for example.
This means that the largest $2$-quotient of $G$ (which is trivial by assumption) is isomorphic to the largest $2$-quotient of $N_G(P)$.
Now $N_G(P)/C_G(P) \le {\rm Aut}(P) \cong {\rm GL}(3,2)$. If $|N_G(P)/C_G(P)|$ is divisible by $7$, then the involutions in $P$ are all conjugate under an element of order $7$. That is what we are trying to prove.
Now $|{\rm GL}(3,2)| = 8 \times 3 \times 7$, and $|N_G(P)/C_G(P)|$ is odd, so the only other options are $|N_G(P)/C_G(P)| = 1$ or $3$. In both cases, the action is reducible with at least one element of order $2$ centralizex by $N_G(P)$, and then $N_G(P)$ has a quotient group of order $2$. Hence so does $G$, contradicting the assumption.