Strongly embedded subgroup

In finite group theory, an area of abstract algebra, a strongly embedded subgroup of a finite group G is a proper subgroup H of even order such that H ∩ H^g has odd order whenever g is not in H. The Bender–Suzuki theorem, proved by (Bender 1971) extending work of Suzuki (1962, 1964), classifies the groups G with a strongly embedded subgroup H. It states that either

1. G has cyclic or generalized quaternion Sylow 2-subgroups and H contains the centralizer of an involution
2. or G/O(G) has a normal subgroup of odd index isomorphic to one of the simple groups PSL₂(q), Sz(q) or PSU₃(q) where q≥4 is a power of 2 and H is O(G)N_G(S) for some Sylow 2-subgroup S.

(Peterfalvi 2000) revised Suzuki's part of the proof.

(Aschbacher 1974) extended Bender's classification to groups with a proper 2-generated core.

References

• Aschbacher, Michael (1974), "Finite groups with a proper 2-generated core", Transactions of the American Mathematical Society 197: 87–112, doi:10.2307/1996929, ISSN 0002-9947 
• Bender, Helmut (1971), "Transitive Gruppen gerader Ordnung, in denen jede Involution genau einen Punkt festläβt", Journal of Algebra 17: 527–554, doi:10.1016/0021-8693(71)90008-1, ISSN 0021-8693 
• Peterfalvi, Thomas (2000), Character theory for the odd order theorem, London Mathematical Society Lecture Note Series, 272, Cambridge University Press, ISBN 978-0-521-64660-4, https://books.google.com/books?isbn=052164660X 
• Suzuki, Michio (1962), "On a class of doubly transitive groups", Annals of Mathematics, Second Series 75: 105–145, doi:10.2307/1970423, ISSN 0003-486X 
• Suzuki, Michio (1964), "On a class of doubly transitive groups. II", Annals of Mathematics, Second Series 79: 514–589, doi:10.2307/1970408, ISSN 0003-486X