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 ∩ Hg 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)NG(S) for some Sylow 2-subgroup S.

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

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

• Aschbacher, Michael (1974), "Finite groups with a proper 2-generated core", Transactions of the American Mathematical Society, 197: 87–112, doi:, ISSN , JSTOR , MR
• Bender, Helmut (1971), "Transitive Gruppen gerader Ordnung, in denen jede Involution genau einen Punkt festläβt", Journal of Algebra, 17: 527–554, doi:, ISSN , MR
• Peterfalvi, Thomas (2000), , London Mathematical Society Lecture Note Series, vol. 272, Cambridge University Press, ISBN 978-0-521-64660-4, MR
• Suzuki, Michio (1962), "On a class of doubly transitive groups", Annals of Mathematics, Second Series, 75: 105–145, doi:, hdl:, ISSN , JSTOR , MR
• Suzuki, Michio (1964), "On a class of doubly transitive groups. II", Annals of Mathematics, Second Series, 79: 514–589, doi:, ISSN , JSTOR , MR