Finite totally k-closed groups1

Dmitry Churikov, Cheryl E. Praeger

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

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 contributionКонечные вполне k-замкнутые группы
Original languageEnglish
Article number20
Pages (from-to)240-245
Number of pages6
JournalTrudy Instituta Matematiki i Mekhaniki UrO RAN
Volume27
Issue number1
DOIs
Publication statusPublished - 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

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