A generalisation of a theorem of Wielandt

TL;DR

This paper extends Wielandt's theorem by showing that subnormality of certain odd order subgroups can be inferred from weaker conditions involving conjugacy classes, and explores properties of subgroups in finite simple groups.

Contribution

It generalizes Wielandt's theorem for odd order nilpotent subgroups and investigates subgroup properties in finite simple groups.

Findings

Subnormality of odd order nilpotent subgroups is implied by conjugacy class conditions.

Existence of cyclic p'-subgroups that do not normalize certain p-subgroups in simple groups.

Extension of subnormality criteria to broader classes of subgroups.

Abstract

In 1974, Helmut Wielandt proved that in a finite group $G$, a subgroup $A$ is subnormal if and only if it is subnormal in every $\seq{A,g}$ for all $g\in G$. In this paper, we prove that the subnormality of an odd order nilpotent subgroup $A$ of $G$ is already guaranteed by a seemingly weaker condition: $A$ is subnormal in $G$ if for every conjugacy class $C$ of $G$ there exists $c\in C$ for which $A$ is subnormal in $\seq{A,c}$. We also prove the following property of finite non-abelian simple groups: if $A$ is a subgroup of odd prime order $p$ in a finite almost simple group $G$, then there exists a cyclic $p'$-subgroup of $F^*(G)$ which does not normalise any non-trivial $p$-subgroup of $G$ that is generated by conjugates of~$A$.