### 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∈VG{0} or v∈V{0} respectively, there exists a homogeneous invariant f∈k[V]G 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 divisible 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.

Original language | English |
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Pages (from-to) | 92-113 |

Number of pages | 22 |

Journal | Journal of Algebra |

Volume | 411 |

Early online date | 13 May 2014 |

DOIs | |

Publication status | Published - 1 Aug 2014 |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Algebra*,

*411*, 92-113. https://doi.org/10.1016/j.jalgebra.2014.03.044

**Zero-separating invariants for finite groups.** / Elmer, Jonathan; Kohls, Martin.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 411, pp. 92-113. https://doi.org/10.1016/j.jalgebra.2014.03.044

}

TY - JOUR

T1 - Zero-separating invariants for finite groups

AU - Elmer, Jonathan

AU - Kohls, Martin

N1 - This paper was prepared during visits of the first author to TU München and the second author to University of Aberdeen. The second of these visits was supported by the Edinburgh Mathematical Society's Research Support Fund. We want to thank Gregor Kemper and the Edinburgh Mathematical Society for making these visits possible. We also thank K. Cziszter for some helpful communication via eMail, and the anonymous referee for useful suggestions.

PY - 2014/8/1

Y1 - 2014/8/1

N2 - 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∈VG{0} or v∈V{0} respectively, there exists a homogeneous invariant f∈k[V]G 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 divisible 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.

AB - 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∈VG{0} or v∈V{0} respectively, there exists a homogeneous invariant f∈k[V]G 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 divisible 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.

UR - http://www.scopus.com/inward/record.url?scp=84899931121&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2014.03.044

DO - 10.1016/j.jalgebra.2014.03.044

M3 - Article

AN - SCOPUS:84899931121

VL - 411

SP - 92

EP - 113

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -