Walter theorem

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title: "Walter theorem" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["theorems-about-finite-groups"] topic_path: "general/theorems-about-finite-groups" source: "https://en.wikipedia.org/wiki/Walter_theorem" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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.

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 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 groups 2G2(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. and later showed that they are all Ree groups, and gave a unified exposition of this result.

References

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