Walter's theorem states that if G is a finite group whose 2-sylow subgroups are abelian, then G/O(G) has a
normal subgroup of odd index that is a product of groups each of which is a 2-group or one of the
simple groups PSL2(q) for q = 2n or q = 3 or 5 mod 8, or the
Janko group J1, or
Ree groups2G2(32n+1). (Here O(G) denotes the unique largest normal subgroup of G of
odd order.)
The original statement of Walter's theorem did not quite identify the Ree groups, but only stated that the corresponding groups have a similar subgroup structure as Ree groups. Thompson (
1967,
1972,
1977) and
Bombieri, Odlyzko & Hunt (1980) later showed that they are all Ree groups, and
Enguehard (1986) gave a unified exposition of this result.