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

TL;DR

This paper investigates embedding properties of $rak{X}$-subgroups within finite groups, providing conditions under which such subgroups can be conjugated into maximal $rak{X}$-subgroups, with examples and new theorems.

Contribution

It introduces new criteria for conjugacy of $rak{X}$-subgroups based on projections and normalizers, extending classical embedding theorems.

Findings

Counterexamples where projections do not imply conjugacy

Proven that normalization of projections ensures conjugacy

Extended results to submaximal $rak{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 1963-64: For a given $\mathfrak{X}$-subgroup $K$ and maximal $\mathfrak{X}$-subgroup $H$, is it possible to see embeddability of $K$ in $H$ (up to conjugacy) by their projections onto 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.