Monolithic Brauer Characters

Abstract

Let $G$ be a group, $p$ be a prime and $P\in \text{Syl}_{p}(G)$. We say that a $p$-Brauer character $\unicode[STIX]{x1D711}$ is monolithic if $G/\ker \unicode[STIX]{x1D711}$ is a monolith. We prove that $P$ is normal in $G$ if and only if $p\nmid \unicode[STIX]{x1D711}(1)$ for each monolithic Brauer character $\unicode[STIX]{x1D711}\in \text{IBr}(G)$. When $G$ is $p$-solvable, we also prove that $P$ is normal in $G$ and $G/P$ is nilpotent if and only if $\unicode[STIX]{x1D711}(1)^{2}$ divides $|G:\ker \unicode[STIX]{x1D711}|$ for all monolithic irreducible $p$-Brauer characters $\unicode[STIX]{x1D711}$ of $G$.

Degrees of Brauer Characters and Normal Sylow -Subgroups

“…The authors with Cossey and Tong-Viet proved in [1] that if ϕ(1) 2 | |G : Ker(ϕ)| for all Brauer characters ϕ ∈ IBr(G), then P G and G/P is nilpotent, where G is a p-solvable group and P is a Sylow p-subgroup of G. In [3], we showed that the full set of irreducible Brauer characters can be replaced by the monolithic irreducible Brauer characters. A group G is said to be a monolith if it contains a unique minimal normal subgroup, and the p-Brauer character ϕ ∈ IBr(G) is called monolithic if G/ Ker(ϕ) is a monolith.…”