In this MathSE question, classification of finite simple groups with Abelian Sylow 2-subgroups, credit is rightly given to John Walter. But in the introduction to his paper, Walter explicitly states that "It seems to be a very difficult problem to show that these are the only examples." Is there a later reference, perhaps earlier than the complete classification theorem, that states that Walter, et al. found them all? Thanks for your help.

## 2 Answers

The remark of Walter in his paper is referring specifically to the groups of Type (3) in his classification, that is, simple groups $S$ such that, for each involution $\tau \in S$, we have $C_S(\tau) \cong \langle \tau \rangle \times {\rm PSL}(2,q)$ with $q \equiv \pm 3 \bmod 8$.

These include the first Janko group $J_1$ (with $q=5$) and the groups of Ree type $^2G_2(q)$ with $q=3^k$ and $k$ odd.

It was quickly proved that any unknown simple group of this type must have similar properties to the groups of Ree type. John Thompson devoted a lot of time trying to prove that there were no further groups of this type, and he eventually reduced it to a problem in algebraic geometry, which was finally settled by Bombieri in 1980 in the paper:

Bombieri, Enrico (1980), appendices by Andrew Odlyzko and D. Hunt, "Thompson's problem ($\sigma^2=3$)", Inventiones Mathematicae, 58 (1): 77–100, doi:10.1007/BF01402275, ISSN 0020-9910, MR 0570875.

Of course "Thompson's Problem ($\sigma^2=3$)" is a strange title for a mathematical paper, but it was solving an important problem! I think Bombieri proved it for sufficiently large $q$, and the appendices of the paper describe computer calculations to settle the remaining small values.

So yes, this was resolved before the complete classification, but not so long before. I remember at the time that people were speculating that this problem might turn out to be the last one to be resolved.

It is described in Gorenstein's book on finite simple groups.