On embedding theorems for ${\mathfrak {X}}$-subgroups

Abstract

Let ${\mathfrak {X}}$ be a class of finite groups closed under subgroups, homomorphic images, and extensions. We study the question which goes back to the lectures of H. Wielandt in 1963–1964: Given an ${\mathfrak {X}}$-subgroup K and a maximal ${\mathfrak {X}}$-subgroup H,  is it possible to detect embeddability of K in H (up to conjugacy) by their projections into the factors of a fixed subnormal series? On the one hand, we construct examples where K has the same projections as some subgroup of H but is not conjugate to any subgroup of H. On the other hand, we prove that if K normalizes the projections of a subgroup H,  then K is conjugate to a subgroup of H even in the more general case when H is a submaximal ${\mathfrak {X}}$-subgroup.