# Metacyclic group

From Encyclopedia of Mathematics

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.

#### Comments

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

#### References

[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: http://encyclopediaofmath.org/index.php?title=Metacyclic_group&oldid=13394

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098