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 ${\displaystyle G}$ having a cyclic normal subgroup ${\displaystyle N}$, such that the quotient ${\displaystyle G/N}$ is also cyclic.

Metacyclic groups are metabelian and supersolvable. In particular, they are solvable.

Definition

A group ${\displaystyle G}$ is metacyclic if it has a normal subgroup ${\displaystyle N}$ such that ${\displaystyle N}$ and ${\displaystyle G/N}$ are both cyclic. ^[1]

In some older books, an inequivalent definition is used: a group ${\displaystyle G}$ is metacyclic if ${\displaystyle [G,G]}$ and ${\displaystyle G/[G,G]}$ are both cyclic. ^[2] 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

1. Kida, Masanari (2012). "On metacyclic extensions". Journal de Théorie des Nombres de Bordeaux. 24 (2): 339–353. ISSN   1246-7405.
2. Hall, Marshall Jr. (1959). The Theory of Groups. New York: Macmillan. MR   0103215.