$(2,3)$-generated groups with small element orders

N. Yang^a, A. S. Mamontov<sup>bc

a</sup> School Sci, Jiangnan Univ., Wuxi, P. R. CHINA
^b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
^c Novosibirsk State University

Abstract: A periodic group is called an $OC_n$-group if the set of its element orders consists of all natural numbers from $1$ to some natural $n$. W. Shi posed the question whether every $OC_n$-group is locally finite. Until now, the case $n=8$ remains open. Here we prove that if a group is generated by an involution and an element of order $3$, and its element orders do not exceed $8$, then it is finite. Thereby we obtain an affirmative answer to Shi's question for $n=8$ for $(2,3)$-generated subgroups.

Keywords: locally finite group, $OC_n$-group, $(2,3)$-generated group, involution.

UDC: 512.542

Received: 02.04.2021
Revised: 18.10.2021

DOI: 10.33048/alglog.2021.60.306

 English version: Algebra and Logic, 2021, 60:3, 217–222

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