On finite totally k-closed groups

Abstract

Let $G$ be a finite group acting faithfully on a finite set $Ω$. For a positive integer $k$, $G$ acts naturally on the Catesian product $Ω^k := Ω\times ...\times Ω$. In this paper, we prove that finite nilpotent group $G$ with $2\nmid |G|$ is a totally $k$-closed group if and only if $G$ is abelian with $n(G)\leq k-1$ or cyclic, where $n(G)$ is the number of invariant factors in the invariant factor decomposition of $G$.

Theorems & Definitions (17)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Theorem 3.1
• proof
• Proposition 3.2