Metacyclic group

In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. That is, it is a group ${\displaystyle G}$ for which there is a short exact sequence

${\displaystyle 1\rightarrow K\rightarrow G\rightarrow H\rightarrow 1,\,}$

where ${\displaystyle H}$ and ${\displaystyle K}$ are cyclic. Equivalently, a metacyclic group is a group ${\displaystyle G}$ having a cyclic normal subgroup ${\displaystyle N}$, such that the quotient ${\displaystyle G/N}$ 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