|
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
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
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
Linking options:
https://www.mathnet.ru/eng/isu908 https://www.mathnet.ru/eng/isu/v21/i4/p442
|
Statistics & downloads: |
Abstract page: | 132 | Full-text PDF : | 54 | References: | 28 |
|