On the modular Plesken Lie algebra

TL;DR

This paper reviews the structure of the Plesken Lie algebra associated with finite groups, discusses known results, and proposes a conjecture for its characteristic p analog, especially when p divides the group order.

Contribution

It provides a comprehensive review of L[G] and introduces a new conjecture on the characteristic p analog L_p[G], expanding understanding of modular Lie algebras.

Findings

L[G] decomposes into simple Lie algebras based on character theory

Known results on L[G] are summarized and reviewed

A conjecture on L_p[G] for prime p dividing |G| is proposed

Abstract

Let G be a finite group. The Plesken Lie algebra L[G] is a subalgebra of the complex group algebra C[G] and admits a direct-sum decomposition into simple Lie algebras based on the ordinary character theory of G. In this paper we review the known results on L[G] and related Lie algebras, as well as introduce a conjecture on a characteristic p analog L_p[G], with a focus on when p divides the order of G.