The structure of groups with cyclic commutator subgroups indecomposable to a subdirect product of groups. (Russian. English summary) Zbl 1539.20008
Summary: 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.
MSC:
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 20F12 | Commutator calculus |
Keywords:
cyclic commutator subgroup; subdirect product of groups; Sylow subgroup; semidirect product of groups; centralizer; group extension; supersolvable groupReferences:
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