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

V. A. Kozlov ; G. N. Titov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 4, pp. 442-447

Résumé

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.