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 Diskr. Mat., 2014, Volume 26, Issue 1, Pages 3–9 (Mi dm1263)

On groups with automorphisms generating recurrent sequences of the maximal period

A. V. Akishin

Moscow State Institute of Radio-Engineering, Electronics and Automation (Technical University)

Abstract: Let $G$ be a finite group and $f$ be an automorphism of the group $G$. The automorphism $f$ specifies a recurrent sequence $\{ a_i \}$ on the group $G$, $i = 0, 1, \ldots$, according to the rule $a_{i+1} = f(a_i)$. If $a_0$ is the initial element of the sequence, then its period does not exceed the number of elements in the group having the same order as the element $a_0$. Thus, it makes sense to formulate the question of whether there exist groups in which such recurrent sequence for a certain automorphism has the maximal period for any initial element. In this paper we introduce the notion of an automorphism of the maximal period and find all Abelian groups and finite groups of odd orders having automorphisms of the maximal period. Also, a number of results for finite groups of even orders are established.

Keywords: finite groups, regular automorphisms, recurrent sequences on groups.

DOI: https://doi.org/10.4213/dm1263

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English version:
Discrete Mathematics and Applications, 2015, 25:4, 187–192

Bibliographic databases:

Document Type: Article
UDC: 519.242.2+519.113.6

Citation: A. V. Akishin, “On groups with automorphisms generating recurrent sequences of the maximal period”, Diskr. Mat., 26:1 (2014), 3–9; Discrete Math. Appl., 25:4 (2015), 187–192

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/dm1263
• https://doi.org/10.4213/dm1263
• http://mi.mathnet.ru/eng/dm/v26/i1/p3

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This publication is cited in the following articles:
1. A. V. Akishin, “On groups of even orders with automorphisms generating recurrent sequences of the maximal period”, Discrete Math. Appl., 25:5 (2015), 253–259
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