# Generalized symmetric group

From Groupprops

This is a variation of symmetric group|Find other variations of symmetric group |

## Contents

## Definition

The **generalized symmetric group** with parameters and , denoted , is defined in the following equivalent ways:

- It is the external wreath product of the cyclic group of order , and the symmetric group (specifically, symmetric group on finite set) of degree , with the natural action of the latter on a set of size .
- It is the external semidirect product of the homocyclic group and the symmetric group of degree , where the latter has a natural action by coordinate permutations.
- It is the subgroup of the general linear group over the field of complex numbers comprising monomial matrices (i.e., matrices where every row has exactly one nonzero entry and every column has exactly one nonzero entry) where all the nonzero entries are roots of unity. Note that the group also has a faithful monomial representation of degree over any field where the polynomial splits completely.
- It is the centralizer inside the symmetric group of degree of a permutation that is a product of disjoint cycles of size each.

## Arithmetic functions

Function | Value | Explanation |
---|---|---|

order |

## Particular cases

### Two very special cases

- In the case , we get the usual symmetric group (specifically, the symmetric group on finite set).
- In the case , we get the signed symmetric group.

## GAP implementation

These groups can be constructed in GAP using the WreathProduct, CyclicGroup and SymmetricGroup functions, as follows:

`WreathProduct(CyclicGroup(m),SymmetricGroup(n))`