On $\mathfrak F$-hypercentral modules and character clusters

TL;DR

This paper investigates the properties of modules over restricted Lie algebras within saturated formations, establishing conditions under which hypercentrality is preserved across modules based on their character clusters.

Contribution

It introduces new criteria linking character clusters and hypercentrality of modules over restricted Lie algebras in saturated formations.

Findings

Hypercentrality of modules is preserved under certain character cluster conditions.

Character clusters determine the transfer of hypercentrality between modules.

Results apply to modules over restricted Lie algebras with specific structural properties.

Abstract

Let $\mathfrak F$ be a saturated formation of soluble Lie algebras over a field $F$ of characteristic $p > 0$ and let ${\mathbb F}_p$ denote the field of $p$ elements. Let $(L,[p])$ be a restricted Lie algebra over $F$ with $z^{$\scriptstyle [p]$}=0$ for all $z$ in the centre of $L$. Let $S \in \mathfrak F$, $S\ne 0$ be a subnormal subalgebra of $L$. Let $V, W$ be $L$-modules. Suppose that the character cluster of $W$ is contained in the set of ${\mathbb F}_p$-linear combinations of the characters in the character cluster of $V$. Suppose that $V$, regarded as $S$-module, is $\mathfrak F$-hypercentral. Then $W$, regarded as $S$-module, is also $\mathfrak F$-hypercentral.