In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. Equivalently, a metacyclic group is a group having a cyclic normal subgroup , such that the quotient is also cyclic.
Metacyclic groups are metabelian and supersolvable. In particular, they are solvable.
Definition
A group is metacyclic if it has a normal subgroup such that and are both cyclic.
In some older books, an inequivalent definition is used: a group is metacyclic if the commutator subgroup and are both cyclic. This is a strictly stronger property than the one used in this article: for example, the quaternion group is not metacyclic by this definition.
Examples Any cyclic group is metacyclic. The direct product or semidirect product of two cyclic groups is metacyclic. These include the dihedral groups and the quasidihedral groups. The dicyclic groups are metacyclic. (Note that a dicyclic group is not necessarily a semidirect product of two cyclic groups.) Every finite group of squarefree order is metacyclic. More generally every Z-group is metacyclic. A Z-group is a finite group whose Sylow subgroups are cyclic.
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