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Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, Volume 21, Issue 4, Pages 442–447
DOI: https://doi.org/10.18500/1816-9791-2021-21-4-442-447
(Mi isu908)
 

Scientific Part
Mathematics

The structure of groups with cyclic commutator subgroups indecomposable to a subdirect product of groups

V. A. Kozlova, G. N. Titovb

a Armavir State Pedagogical University, 159 Rosa Luxemburg St., Armavir 352901, Russia
b Kuban State University, 149 Stavropolskaya St., Krasnodar 350040, Russia
References:
Abstract: 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.
Key words: group, cyclic commutator subgroup, subdirect product of groups, Sylow subgroup, semidirect product of groups, centralizer, group extension, supersolvable group.
Received: 15.03.2021
Accepted: 03.08.2021
Bibliographic databases:
Document Type: Article
UDC: 512.54
Language: Russian
Citation: V. A. Kozlov, G. N. Titov, “The structure of groups with cyclic commutator subgroups indecomposable to a subdirect product of groups”, Izv. Saratov Univ. Math. Mech. Inform., 21:4 (2021), 442–447
Citation in format AMSBIB
\Bibitem{KozTit21}
\by V.~A.~Kozlov, G.~N.~Titov
\paper The structure of groups with cyclic commutator subgroups indecomposable to a subdirect product of groups
\jour Izv. Saratov Univ. Math. Mech. Inform.
\yr 2021
\vol 21
\issue 4
\pages 442--447
\mathnet{http://mi.mathnet.ru/isu908}
\crossref{https://doi.org/10.18500/1816-9791-2021-21-4-442-447}
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    Известия Саратовского университета. Новая серия. Серия Математика. Механика. Информатика
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