Metacyclic group

Template:Short description 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

$1\to K\to G\to H\to 1,$

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

• Template:Springer

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