Metacyclic group

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.

References