Characterization of $A_5$ by the Centralizer of an Involution

Solution 1:

To me the proof given in your answer looks fine.

Regarding Fowler, I have access to Fowler's 1952 thesis, and I cannot find the result you mention. The main result of his thesis is the following:

Theorem: Let $G$ be a finite group of order $2^a n$, where $n$ is odd. If $G$ contains no element with order $2p$ for $p$ an odd prime, and if the $2$-Sylow subgroups $P$ of $G$ are abelian, then one of the following must hold:

(a) $P$ is a normal subgroup of $G$.

(b) $P$ is cyclic and $G$ contains a normal subgroup of index $2^a$.

(c) $G$ is isomorphic to $\operatorname{SL}(2, 2^a)$.

One part of the Brauer-Suzuki-Wall theorem is a consequence of Fowler's result.