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 extending work of,  classifies the groups G with a strongly embedded subgroup H. It states that either revised Suzuki's part of the proof.
 * 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.

extended Bender's classification to groups with a proper 2-generated core.