In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. That is, it is a group G for which there is a short exact sequence
where H and K are cyclic. Equivalently, a metacyclic group is a group G having a cyclic normal subgroup N, such that the quotient G/N is also cyclic.
Properties
Metacyclic groups are both supersolvable and metabelian.
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 group whose Sylow subgroups are cyclic.
References
- A. L. Shmel'kin (2001) [1994], "Metacyclic group", Encyclopedia of Mathematics, EMS Press