Metacyclic group

Short description: Extension of a cyclic group by a cyclic group

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

[math]\displaystyle{ 1 \rightarrow K \rightarrow G \rightarrow H \rightarrow 1,\, }[/math]

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

• Hazewinkel, Michiel, ed. (2001), "Metacyclic group", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=M/m063550