finite group theory In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

, an area of

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

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

proper subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

''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 \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...

of an

involution Involution may refer to: Mathematics * Involution (mathematics), a function that is its own inverse * Involution algebra, a *-algebra: a type of algebraic structure * Involute, a construction in the differential geometry of curves * Exponentiati ...

# 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 ...

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 d ...

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