metacyclic group
Definition
A metacyclic group^{} is a group $G$ that possesses a normal subgroup^{} $N$ such that $N$ and $G/N$ are both cyclic.
Examples

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All cyclic groups^{}, and direct products^{} of two cyclic groups.

•
All dihedral groups^{} (including the infinite dihedral group).

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All finite groups^{} whose Sylow subgroups are cyclic (and so, in particular, all finite groups of squarefree (http://planetmath.org/SquareFreeNumber) order).
Properties
Subgroups^{} (http://planetmath.org/Subgroup) and quotients^{} (http://planetmath.org/QuotientGroup) of metacyclic groups are also metacyclic.
Metacyclic groups are obviously supersolvable, with Hirsch length at most $2$.
Title  metacyclic group 

Canonical name  MetacyclicGroup 
Date of creation  20130322 15:36:39 
Last modified on  20130322 15:36:39 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  5 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 20F16 
Defines  metacyclic 