Gorenstein–Walter theorem

In mathematics, the Gorenstein–Walter theorem, proved by Gorenstein and Walter (1965a, 1965b, 1965c), states that if a finite group G has a dihedral Sylow 2-subgroup, and O(G) is the maximal normal subgroup of odd order, then G/O(G) is isomorphic to a 2-group, or the alternating group A₇, or a subgroup of PΓL₂(q) containing PSL₂(q) for q an odd prime power. Note that A₅ ≈ PSL₂(4) ≈ PSL₂(5) and A₆ ≈ PSL₂(9).

References

• Gorenstein, D.; Walter, John H. (1965a), "The characterization of finite groups with dihedral Sylow 2-subgroups. I", Journal of Algebra, 2 (1): 85–151, doi:10.1016/0021-8693(65)90027-X, ISSN 0021-8693, MR 0177032
• Gorenstein, D.; Walter, John H. (1965b), "The characterization of finite groups with dihedral Sylow 2-subgroups. II", Journal of Algebra, 2 (2): 218–270, doi:10.1016/0021-8693(65)90019-0, ISSN 0021-8693, MR 0177032
• Gorenstein, D.; Walter, John H. (1965c), "The characterization of finite groups with dihedral Sylow 2-subgroups. III", Journal of Algebra, 2 (3): 354–393, doi:10.1016/0021-8693(65)90015-3, ISSN 0021-8693, MR 0190220