Zero-separating invariants for finite groups

TL;DR

This paper investigates zero-separating invariants for finite groups over fields of characteristic p, establishing exact values and bounds for these invariants in the modular case, extending known non-modular results.

Contribution

It provides new formulas and bounds for the invariants elta(G) and ap(G) in the modular setting, generalizing classical results.

Findings

elta(G) equals the order of a Sylow p-subgroup P.

ap(G) is at least the order of the normalizer of P modulo P when the quotient is cyclic.

Bounds on ap(G) are given for p-nilpotent groups with non-normal Sylow p-subgroups.

Abstract

We fix a field $\kk$ of characteristic $p$. For a finite group $G$ denote by $\delta(G)$ and $\sigma(G)$ respectively the minimal number $d$, such that for any finite dimensional representation $V$ of $G$ over $\kk$ and any $v\in V^{G}\setminus\{0\}$ or $v\in V\setminus\{0\}$ respectively, there exists a homogeneous invariant $f\in\kk[V]^{G}$ of positive degree at most $d$ such that $f(v)\ne 0$. Let $P$ be a Sylow-$p$-subgroup of $G$ (which we take to be trivial if the group order is not divisble by $p$). We show that $\delta(G)=|P|$. If $N_{G}(P)/P$ is cyclic, we show $\sigma(G)\ge|N_{G}(P)|$. If $G$ is $p$-nilpotent and $P$ is not normal in $G$, we show $\sigma(G)\le \frac{|G|}{l}$, where $l$ is the smallest prime divisor of $|G|$. These results extend known results in the non-modular case to the modular case.