finite group theory Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

, an area of

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...

, a strongly embedded subgroup of a finite group ''G'' is a

proper subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...

''H'' of even order such that ''H'' ∩ ''H''^''g'' has odd order whenever ''g'' is not in ''H''. The Bender–Suzuki theorem, proved by extending work of , classifies the groups ''G'' with a strongly embedded subgroup ''H''. It states that either # ''G'' has cyclic or generalized quaternion Sylow 2-subgroups and ''H'' contains the

centralizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', ...

of an

involution Involution may refer to: * Involute, a construction in the differential geometry of curves * ''Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour input ...

# or ''G''/''O''(''G'') has a

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...

of odd index isomorphic to one of the

simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The da ...

s 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''. revised Suzuki's part of the proof. extended Bender's classification to groups with a proper 2-generated core.

References

* * * * *{{Citation , last1=Suzuki , first1=Michio , author1-link=Michio Suzuki (mathematician) , title=On a class of doubly transitive groups. II , jstor=1970408 , mr=0162840 , year=1964 , journal=

Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...

, series=Second Series , issn=0003-486X , volume=79 , pages=514–589 , doi=10.2307/1970408 Finite groups