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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.
Original language | English |
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Pages (from-to) | 240-245 |
Number of pages | 6 |
Journal | Trudy Instituta Matematiki i Mekhaniki |
Volume | 27 |
Issue number | 1 |
DOIs | |
Publication status | Published - 2021 |
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Dive into the research topics of 'Finite totally k-closed groups1'. Together they form a unique fingerprint.Projects
- 1 Finished
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Complexity of group algorithms and statistical fingerprints of groups
Praeger, C. (Investigator 01) & Niemeyer, A. (Investigator 02)
ARC Australian Research Council
21/02/19 → 31/12/22
Project: Research