In mathematics, the Walter theorem, proved by , describes the finite groups whose Sylow 2-subgroup is abelian. used Bender's method

Bender's method

In group theory, Bender's method is a method introduced by for simplifying the local group theoretic analysis of the odd order theorem. Shortly afterwards he used it to simplify Walter's analysis of groups with abelian Sylow 2-subgroups , and Gorenstein and Walter's classification of groups with...

 to give a simpler proof.

Statement

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 of one of the simple groups PSL₂(q) for q = 2n or q = 3 or 5 mod 8, or the Janko group J1

Janko group J1

In mathematics, the smallest Janko group, J1, is a simple sporadic group of order 175560. It was originally described by Zvonimir Janko and was the first sporadic group to be found since the discovery of the Mathieu groups in the 19th century...

, or Ree groups ²G₂(3²ⁿ⁺¹).

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.

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