Metacyclic group
Extension of a cyclic group by a cyclic group
From Wikipedia, the free encyclopedia
In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. Equivalently, a metacyclic group is a group having a cyclic normal subgroup , such that the quotient is also cyclic.
Metacyclic groups are metabelian and supersolvable. In particular, they are solvable.
Definition
A group is metacyclic if it has a normal subgroup such that and are both cyclic.[1]
In some older books, an inequivalent definition is used: a group is metacyclic if the commutator subgroup and 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.
![{\displaystyle [G,G]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/6ddf7a724a331d1e12ffa6571ba246ebf08f1335)
![{\displaystyle G/[G,G]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/169489000a5a3370a8d0a56d35924011e53b6ab1)
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.
