In mathematics, the Walter theorem, proved by , describes the finite groups whose Sylow 2-subgroup is abelian. used Bender's method to give a simpler proof.
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 PSL<sub>2</sub>(q) for q = 2<sup>n</sup> or q = 3 or 5 mod 8, or the Janko group J1, or Ree groups <sup>2</sup>G<sub>2</sub>(3<sup>2n+1</sup>). (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. and later showed that they are all Ree groups, and gave a unified exposition of this result.