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# 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 that either

*G*has cyclic or generalized quaternion Sylow 2-subgroups and*H*contains the centralizer of an involution- or
*G*/*O*(*G*) has a normal subgroup of odd index isomorphic to one of the simple groups PSL_{2}(*q*), Sz(*q*) or PSU_{3}(*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, part II) 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, ISSN 0002-9947, JSTOR 1996929, MR 0364427

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, MR 0288172

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, MR 1747393

Suzuki, Michio (1962), "On a class of doubly transitive groups", Annals of Mathematics. Second Series 75: 105–145, ISSN 0003-486X, JSTOR 1970423, MR 0136646

Suzuki, Michio (1964), "On a class of doubly transitive groups. II", Annals of Mathematics. Second Series 79: 514–589, ISSN 0003-486X, JSTOR 1970408, MR 0162840

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