### 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 |

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Journal of Algebra*,

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