Two Criteria for Nonsimplicity of a Group Possessing a Strongly Embedded Subgroup and a Finite Involution

Abstract

A proper subgroup H of a group G is said to be strongly embedded if 2 ∈π(H) and 2∉π(H∩Hg) (for all \(g \in G\backslash H\)). An involution i of G is said to be finite if \(\left| {ii^g } \right| < \infty\) (for all g∈ G). As is known, the structure of a (locally) finite group possessing a strongly embedded subgroup is determined by the theorems of Burnside and Brauer--Suzuki, provided that the Sylow 2-subgroup contains a unique involution. In this paper, sufficient conditions for the equality m2(G)= 1 are established, and two analogs of the Burnside and Brauer—Suzuki theorems for infinite groups G possessing a strongly embedded subgroup and a finite involution are given.