metacyclic group
Definition
A metacyclic group is a group that possesses a normal subgroup
such that and are both cyclic.
Examples
- •
All cyclic groups
, and direct products
of two cyclic groups.
- •
All dihedral groups
(including the infinite dihedral group).
- •
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 .