Zero-separating invariants for finite groups

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.

16 pages

References in corpus (2)

• Degree bounds for separating invariants
• Invariants of the dihedral group $D_{2p}$ in characteristic two

Cited by in corpus (7)

• Estimation under group actions: recovering orbits from invariants
• Separating invariants over finite fields
• Separating invariants for 2x2 matrices
• Separating invariants for multisymmetric polynomials
• Degree bounds for fields of rational invariants of $\mathbb{Z}/p\mathbb{Z}$ and other finite groups
• Degree of reductivity of a modular representation
• On separating fixed points from zero by invariants