Geodesic
Finite totally $k$-closed groups
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 1, pp. 240-245 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 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 finite 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.
Keywords: permutation group; $k$-closure; totally $k$-closed group. 
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D. V. Churikov; Ch. E. Praeger. Finite totally $k$-closed groups. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 1, pp. 240-245. http://geodesic.mathdoc.fr/item/TIMM_2021_27_1_a20/

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