Layer-finiteness of some groups

Vladimir I. Senashov

The Bulletin of Irkutsk State University. Series Mathematics, Tome 48 (2024), pp. 145-151

Résumé

Infinite groups with finiteness conditions for an infinite system of subgroups are studied. Groups with a condition: the normalizer of any non-trivial finite subgroup is a layer-finite group or the normalizer of any non-trivial finite subgroup has a layer-finite periodic part are studied for beginning in the locally finite class of group, then in the class of periodic groups of Shunkov and finally in the class of Shunkov groups which are contain a strongly embedded subgroup with an almost layer-finite periodic part. The group $G$ is called the Shunkov group if for any prime $p$ and for every finite subgroups $H$ from $G$ any two conjugate elements of order $p$ from the factor-group $N_G(H)/H$ generate a finite subgroup. Results for almost layer-finite groups and groups with almost layer-finite periodic part are transferred to layer-finite groups and groups with layer-finite periodic part. New characterizations of layer-finite groups and groups with layer-finite periodic part are obtained.

Détail

Keywords: periodic part, layer-finite group, almost layer-finite group, Shunkov group.

Bibliographie

[1] Durakov B.E., “On groups saturated by finite Frobenius groups with complements of even orders”, Algebra and Logic, 60 (2021), 560–574 (in Russian) | DOI

[2] Durakov B.E., Sozutov A.I., “On groups with involutions saturated by finite frobenius groups”, Sib. Math. J., 63 (2022), 1256–1265 (in Russian) | DOI

[3] Kurosh A.G., Group theory, Science Publ, M., 1967, 648 pp. | DOI | MR

[4] Senashov V.I., Shunkov V.P., “Almost layer finiteness of the periodic part of a group without involutions”, Discrete Math., 15:3 (2003), 91–104 (in Russian) | DOI | Zbl

[5] Senashov V.I., “Structure of an infinite Sylow subgroup in some periodic Shunkov groups”, Discrete Math., 14 (2002), 133–152 (in Russian) | DOI | Zbl

[6] Senashov V.I., “Characterization of Shunkov groups”, Ukranian math. journal, 60 (2008), 1110–1118 (in Russian) | MR | Zbl

[7] Senashov V.I., “On groups with a strongly embedded subgroup that has an almost layer-finite periodic part”, Ukranian math. journal, 64 (2012), 384–391 (in Russian) | Zbl

[8] Chernikov S.N., “On the theory of infinite p-groups”, Dokl. Academy of Sciences of the USSR, 1945, 71–74 (in Russian) | Zbl

[9] Chernikov S.N., Groups with given properties of systems of subgroups, Nauka Publ, M., 384 pp.

[10] Shlepkin A.A., “On Shunkov groups saturated by general linear groups”, Sib. Math. J., 57 (2016), 222–235 (in Russian) | DOI | Zbl

[11] Shlepkin A.A., “On one sufficient condition for the existence of a periodic part in the Shunkov group”, The Bulletin of Irkutsk State University. Series Mathematics, 22 (2017), 90–105 | DOI | MR | Zbl

[12] Shlepkin A.A., “On a periodic part of a Shunkov group saturated with wreathed groups”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 24, 2018, 281–285 (in Russian) | DOI | MR

[13] Shlepkin A.A., “On the periodic part of the Shunkov group saturated with linear groups of degree 2 over finite fields of even characteristic”, Chebyshevskii Sbornik, 20 (2019), 399–407 (in Russian) | DOI | MR | Zbl

[14] Shlepkin A.A., “On Sylow 2-subgroups of Shunkov groups saturated with $L_3(2^m)$ groups”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 25, 2019, 275–282 (in Russian) | DOI | MR

[15] Shmidt O.Yu., “Groups all of whose subgroups are special”, Math. Sbornik, 1924, no. 31, 366–372 (in Russian)

[16] Shunkov V.P., “On Chernikov $p$-groups”, All-Union Algebraic Symposium, Abstracts of reports, 1975, 189–190 (in Russian)