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

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.

2 pages

References in corpus (2)

• Character clusters for Lie algebra modules over a field of non-zero characteristic
• Induced modules for modular Lie algebras