Finite totally $k$-closed groups

TL;DR

This paper characterizes totally $k$-closed groups, showing all abelian groups are totally $(n(G)+1)$-closed but not $n(G)$-closed, and demonstrates the existence of infinitely many finite abelian $p$-groups with specific total $k$-closure properties.

Contribution

It establishes a precise relationship between the invariant factors of abelian groups and their total $k$-closure properties, extending previous results for $k=2$.

Findings

All abelian groups are totally $(n(G)+1)$-closed.

Existence of infinitely many finite abelian $p$-groups that are totally $k$-closed but not $(k-1)$-closed.

Provides open questions on total $k$-closure properties.

Abstract

For a positive integer $k$, a group $G$ is said to be totally $k$-closed if in each of its faithful permutation representations, say on a set $\Omega$, $G$ is the largest subgroup of $\operatorname{Sym}(\Omega)$ which leaves invariant each of the $G$-orbits in the induced action on $\Omega\times\dots\times \Omega=\Omega^k$. We prove that every abelian group $G$ is totally $(n(G)+1)$-closed, but is not totally $n(G)$-closed, where $n(G)$ is the number of invariant factors in the invariant factor decomposition of $G$. In particular, we prove that for each $k\geq2$ and each prime $p$, there are infinitely many finite abelian $p$-groups which are totally $k$-closed but not totally $(k-1)$-closed. This result in the special case $k=2$ is due to Abdollahi and Arezoomand. We pose several open questions about total $k$-closure.