Geodesic
$(2,3)$-generated groups with small element orders
Algebra i logika, Tome 60 (2021) no. 3, pp. 327-334
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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, $(2,3)$-generated group, involution.
Mots-clés : $OC_n$-group 
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N. Yang; A. S. Mamontov. $(2,3)$-generated groups with small element orders. Algebra i logika, Tome 60 (2021) no. 3, pp. 327-334. http://geodesic.mathdoc.fr/item/AL_2021_60_3_a5/

[1] R. Brandl, W. Shi, “Finite groups whose element orders are consecutive integers”, J. Algebra, 143:2 (1991), 388–400 | DOI | Zbl

[2] Unsolved problems in group theory. The Kourovka notebook, 19, Sobolev Institute of Mathematics, Novosibirsk, 2018 http://www.math.nsc.ru/ãlglog/19tkt.pdf

[3] I. N. Sanov, “Reshenie problemy Bernsaida dlya pokazatelya 4”, Uch. zapiski LGU. Ser. matem., 10 (1940), 166–170

[4] V. D. Mazurov, “O gruppakh perioda 60 s zadannymi poryadkami elementov”, Algebra i logika, 39:3 (2000), 329–346 | Zbl

[5] D. V. Lytkina, V. D. Mazurov, A. S. Mamontov, E. Yabara, “Gruppy, poryadki elementov kotorykh ne prevoskhodyat 6”, Algebra i logika, 53:5 (2014), 570–586 | Zbl

[6] A. S. Mamontov, “O periodicheskikh gruppakh, izospektralnykh $A_7$”, Sib. matem. zh., 61:1 (2020), 137–147 | Zbl

[7] A. S. Mamontov, E. Yabara, “O periodicheskikh gruppakh, izospektralnykh $A_7$. II”, Sib. matem. zh., 61:6 (2020), 1366–1376 | Zbl

[8] A. S. Mamontov, E. Yabara, “Raspoznavanie $A_7$ po mnozhestvu poryadkov elementov”, Sib. matem. zh., 62:1 (2021), 117–130 | Zbl

[9] GAP — Groups, Algorithms, Programming — A System for Computational Discrete Algebra, vers. 4.11.1, , The GAP Group, 2021 https://www.gap-system.org

[10] H. S. M. Coxeter, W. O. J. Moser, Generators and relations for discrete groups, Ergeb. Math. Grenzgeb., 14, 4th ed., Springer-Verlag, Berlin-Heidelberg-New York, 1980 | Zbl