Walter theorem

In mathematics, the Walter theorem, proved by John H. Walter (1967, 1969), describes the finite groups whose Sylow 2-subgroup is abelian. Bender (1970) 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. 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.

References

Bender, Helmut (1970), "On groups with abelian Sylow 2-subgroups", Mathematische Zeitschrift, 117: 164–176, doi:10.1007/BF01109839, ISSN 0025-5874, MR 0288180
Bombieri, Enrico; Odlyzko, Andrew; Hunt, D. (1980), "Thompson's problem (σ²=3)", Inventiones Mathematicae, 58 (1): 77–100, doi:10.1007/BF01402275, ISSN 0020-9910, MR 0570875
Enguehard, Michel (1986), "Caractérisation des groupes de Ree", Astérisque (142): 49–139, ISSN 0303-1179, MR 0873958
Thompson, John G. (1967), "Toward a characterization of E₂^*(q)", Journal of Algebra, 7: 406–414, doi:10.1016/0021-8693(67)90080-4, ISSN 0021-8693, MR 0223448
Thompson, John G. (1972), "Toward a characterization of E₂^*(q). II", Journal of Algebra, 20: 610–621, doi:10.1016/0021-8693(72)90074-9, ISSN 0021-8693, MR 0313377
Thompson, John G. (1977), "Toward a characterization of E₂^*(q). III", Journal of Algebra, 49 (1): 162–166, doi:10.1016/0021-8693(77)90276-9, ISSN 0021-8693, MR 0453858
Walter, John H. (1967), "Finite groups with abelian Sylow 2-subgroups of order 8", Inventiones Mathematicae, 2: 332–376, doi:10.1007/BF01428899, ISSN 0020-9910, MR 0218445
Walter, John H. (1969), "The characterization of finite groups with abelian Sylow 2-subgroups.", Annals of Mathematics, Second Series, 89: 405–514, doi:10.2307/1970648, ISSN 0003-486X, JSTOR 1970648, MR 0249504