Zero-separating invariants for finite groups

Elmer, Jonathan ORCID: https://orcid.org/0000-0001-5296-1987 and Kohls, Martin (2014) Zero-separating invariants for finite groups. Journal of Algebra, 411 . pp. 92-113. ISSN 0021-8693 [Article] (doi:10.1016/j.jalgebra.2014.03.044)

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Abstract

We fix a field k of characteristic p. For a finite group G denote by δ(G) and σ(G) respectively the minimal number d, such that for every finite dimensional representation V of G over k and every v ∈ V^G \ {0} or v ∈ V \ {0} respectively, there exists a homogeneous invariant f of positive degree at most d such that f(v) = 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 δ(G) = |P|. If N_G(P)/P is cyclic, we show σ(G) ≥ |N_G(P)|. If G is p-nilpotent and P is not normal in G, we show σ(G) ≤ |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.

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