Gorenstein–Harada theorem

In mathematical finite group theory, the Gorenstein–Harada theorem, proved by Gorenstein and Harada (1973, 1974) in a 464 page paper, classifies the simple finite groups of sectional 2-rank at most 4. It is part of the classification of finite simple groups.

Finite simple groups of section 2 rank at least 5 have Sylow 2-subgroups with a self-centralizing normal subgroup of rank at least 3, which implies that they have to be of either component type or of characteristic 2 type. Therefore the Gorenstein–Harada theorem splits the problem of classifying finite simple groups into these two subcases.
References

Gorenstein, D.; Harada, Koichiro (1973), "Finite groups of sectional 2-rank at most 4", in Gagen, Terrence; Hale, Mark P. Jr.; Shult, Ernest E., Finite groups '72. Proceedings of the Gainesville Conference on Finite Groups, March 23-24, 1972, North-Holland Math. Studies 7, Amsterdam: North-Holland, pp. 57–67, ISBN 978-0-444-10451-9, MR 0352243
Gorenstein, D.; Harada, Koichiro (1974), Finite groups whose 2-subgroups are generated by at most 4 elements, Memoirs of the American Mathematical Society 147, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1847-3, MR 0367048

Mathematics Encyclopedia