Metacyclic 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
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.
gollark: Well, it won't ping me with my reminders for a week.
gollark: Yes. You pinged me, thus I was pinged.
gollark: ++remind 1w \<@160279332454006795>
gollark: ++remind 1mo <@!160279332454006795>
gollark: ++help
References
- A. L. Shmel'kin (2001) [1994], "Metacyclic group", Encyclopedia of Mathematics, EMS Press
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