For citation:
Kozlov V. A., Titov G. N. The structure of groups with cyclic commutator subgroups indecomposable to a subdirect product of groups. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 4, pp. 442-447. DOI: 10.18500/1816-9791-2021-21-4-442-447, EDN: UJZYCX
The structure of groups with cyclic commutator subgroups indecomposable to a subdirect product of groups
The article studies finite groups indecomposable to subdirect product of groups (subdirectly irreducible groups), commutator subgroups of which are cyclic subgroups. The article proves that extensions of a primary cyclic group by any subgroup of its automorphisms completely describe the structure of non-primary finite subdirectly irreducible groups with a cyclic commutator subgroup. The following theorem is the main result of this article: a finite non-primary group is subdirectly irreducible with a cyclic commutator subgroup if and only if for some prime number $p\geq 3$ it contains a non-trivial normal cyclic $p$-subgroup that coincides with its centralizer in the group. In addition, it is shown that the requirement of non-primality in the statement of the theorem is essential.
- Gorchakov Yu. M. Teoriya grupp [The Theory of Groups]. Tver, TSU, 1998. 112 p. (in Russian).
- Gorchakov Yu. M. Gruppy s konechnymi klassami sopryazhennykh elementov [Groups with Finite Conjugacy Classes]. Moscow, Nauka, 1978. 120 p. (in Russian).
- Kargapolov M. I., Merzljakov Ju. I. Fundamentals of the Theory of Groups. New York, Springer-Verlag, 1979. 203 p. (Russ. ed.: Moscow, Nauka, 1982. 288 p.).
- Cheng Y. On finite p-groups with cyclic commutator subgroup. Archiv der Mathematik, 1982, vol. 39, iss. 4, pp. 295–298. https://doi.org/10.1007/BF01899434
- Dark R. S., Newell M. L. On conditions for commutators to form a subgroup. Journal of the London Mathematical Society, 1978, vol. s2-17, iss. 2, pp. 251–162. https://doi.org/ 10.1112/jlms/s2-17.2.251
- Leong Y. K. Odd order nilpotent groups of class two with cyclic center. Journal of the Australian Mathematical Society, 1974, vol. 17, iss. 2, pp. 142–153. https://doi.org/10. 1017/S1446788700016724
- Leong Y. K. Finite 2-groups of class two with cyclic center. Journal of the Australian Mathematical Society, 1979, vol. 27, iss. 2, pp. 125–140. https://doi.org/10.1017/S1446788700012052
- Miech R. J. On p-groups with cyclic commutator subgroup. Journal of the Australian Mathematical Society, 1975, vol. 20, iss. 2, pp. 178–198. https://doi.org/10.1017/ S1446788700020486
- Finogenov A. A. Finite p-groups with cyclic commutator subgroup and cyclic center. Mathematical Notes, 1998, vol. 63, iss. 6, pp. 802–812. https://doi/10.1007/BF02312775
- Skuratovskii R. V. Commutator subgroup of Sylow 2-subgroups of alternating group and the commutator width in the wreath product. Buletinul Academiei de Stiinte a Republicii Moldova. Matematica, 2020, iss. 1, pp. 3–16.
- Hall M. Teoriya grupp [The Theory of Groups]. Moscow, Inostrannaya literatura, 1962. 468 p. (in Russian).
- Chernikov S. N. Gruppy s zadannymi svoystvami sistemy podgrupp [Groups with Given Properties of a System of Subgroups]. Moscow, Nauka, 1980. 384 p. (in Russian).
- Shemetkov L. A. Formatsii konechnykh grupp [Formations of Finite Groups]. Moscow, Nauka, 1978. 272 p. (in Russian).
- 1365 reads