Metacyclic group

From Encyclopedia of Mathematics
Jump to: navigation, search

A group having a cyclic normal subgroup such that the quotient group by this normal subgroup is also cyclic (cf. Cyclic group). Every finite group of square-free order (i.e. the order is not divisible by the square of a natural number) is metacyclic. Polycyclic groups (cf. Polycyclic group) are a generalization of metacyclic groups.


Sometimes, the term metacyclic is reserved for the more special class of groups whose derived group and derived quotient group are both cyclic.


[a1] M. Hall jr., "The theory of groups" , Macmillan (1959)
How to Cite This Entry:
Metacyclic group. A.L. Shmel'kin (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098