## 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 , G is the largest subgroup of Sym() which leaves invariant each of the G-orbits in the induced action on × × = 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 ≥ 2 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.

Translated title of the contribution | Конечные вполне k-замкнутые группы |
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Original language | English |

Article number | 20 |

Pages (from-to) | 240-245 |

Number of pages | 6 |

Journal | Trudy Instituta Matematiki i Mekhaniki UrO RAN |

Volume | 27 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2021 |

## Keywords

- K-closure
- Permutation group
- Totally k-closed group

## OECD FOS+WOS

- 1.01 MATHEMATICS

## State classification of scientific and technological information

- 27 MATHEMATICS