Gorenstein–Walter theorem
In algebra, the Gorenstein–Walter theorem, proved by Gorenstein and Walter,[1][2][3] 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).
Notes
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. 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. 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. MR 0190220.