On groups of even orders with automorphisms generating recurrent sequences of the maximal period

A. V. Akishin

Moscow Institute of Radio-Engineering, Electronics and Automation

Abstract: Let $G$ be a finite group and $f$ be an automorphism of the group $G$. The automorphism $f$ specifies a recurrent sequence $\left\{ a_i \right\}_0^\infty$ on the group $G$ by the rule $a_{i+1} = f(a_i)$. If $a_0$ is the initial element of the sequence, then the period of the sequence does not exceed the number of elements having the same order as $a_0$. Thus, it is interesting to find out whether there exist groups having automorphisms generating sequences with the largest possible period for any initial element. This work continues the study of groups possessing automorphisms of the maximal period. Earlier, the case of groups of odd orders was examined. It was established that such groups are necessarily Abelian, and their structure was completely described. This paper considers groups of even orders and completes the description of finite groups possessing automorphisms of the maximal period.

Keywords: groups, automorphisms, recurrence sequences.

UDC: 512.542+512.74

Received: 23.05.2014

DOI: 10.4213/dm1300

 English version: Discrete Mathematics and Applications, 2015, 25:5, 253–259

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