Classification of finite simple groups with abelian Sylow 2-subgroups

Question

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.

Answer 1

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.

Answer 2

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