# OpenMath Content Dictionary: permgp2

Canonical URL:
http://www.openmath.org/cd/permgp2.ocd
CD Base:
http://www.openmath.org/cd
CD File:
permgp2.ocd
CD as XML Encoded OpenMath:
permgp2.omcd
Defines:
alternating_group, cyclic_group, dihedral_group, quaternion_group, symmetric_group, vierer_group
Date:
2004-06-01
Version:
1 (Revision 1)
Review Date:
Status:
experimental

A CD of functions for permutation groups. Primarily for defining the best known permutation groups.

     Built by Arjeh M. Cohen 2003-02-16.


## symmetric_group

Description:

This symbol represents a unary function. Its argument is either a positive integer or a set. When evaluated on a set, it represents the permutation group of all permutations of that set. When evaluated on a positive integer n, it represents the permutation group of all permutations of the set {1,..., n}.

Example:
The permutation group generated by (1,2) and (2,3) is equal to the symmetric group on {1,2,3}.
$\mathrm{group}\left(\mathrm{right_compose},\mathrm{permutation}\left(\mathrm{cycle}\left(1,2\right)\right),\mathrm{permutation}\left(\mathrm{cycle}\left(2,3\right)\right)\right)={S}_{3}$
Signatures:
sts

 [Next: alternating_group] [Last: vierer_group] [Top]

## alternating_group

Description:

This symbol represents a unary function. Its argument is either a positive integer or a set. When evaluated on a set, it represents the permutation group of all even permutations of that set. When evaluated on a positive integer n, it represents the permutation group of all even permutations of the set {1,..., n}.

Example:
The permutation group generated by (1,2,3) and (3,4,5) is equal to the alternating group on {1,2,3,4,5}.
$\mathrm{group}\left(\mathrm{right_compose},\mathrm{permutation}\left(\mathrm{cycle}\left(1,2,3\right)\right),\mathrm{permutation}\left(\mathrm{cycle}\left(3,4,5\right)\right)\right)={\mathrm{Alt}}_{5}$
Signatures:
sts

 [Next: cyclic_group] [Previous: symmetric_group] [Top]

## cyclic_group

Description:

This symbol represents a unary function whose argument should be a positive integer. When evaluated at the integer n, it represents the permutation group generated by the permutation (1,2,...,n).

Signatures:
sts

 [Next: dihedral_group] [Previous: alternating_group] [Top]

## dihedral_group

Description:

This symbol represents a unary function whose argument should be a positive integer. When evaluated at the integer n, it represents the dihedral group of all 2n permutations of {1,2,...,n} preserving the n-gon 1,2,...,n.

Commented Mathematical property (CMP):
The group is generated by the permutations (1,2,...,n) and (1,n)(2,n-1)(3,n-3) ....(n/2-1/2,n/2+1/2) if n is odd and by the permutations (1,2,...,n) and (1,n)(2,n-1)(3,n-3) ....(n/2-1,n/2+1) if n is odd.
Example:
The dihedral group on 3 (letters) coincides with the symmetric group on 3 (letters).
${D}_{3}={S}_{3}$
Signatures:
sts

 [Next: quaternion_group] [Previous: cyclic_group] [Top]

## quaternion_group

Description:

This symbol represents the quaternion group of order 8, viewed as a permutation group by means of the regular representation (multiplication from the right). It is generated by (1,2,3,4)(5,8,6,7) and (1,5,2,6)(3,7,4,8). (In the usual notation, the 8 elements are 1, -1, i, -i, j, -j, k, -k.)

Formal Mathematical property (FMP):
$\mathrm{group}\left(\mathrm{right_compose},\mathrm{permutation}\left(\mathrm{cycle}\left(1,3,2,4\right),\mathrm{cycle}\left(5,8,6,7\right)\right),\mathrm{permutation}\left(\mathrm{cycle}\left(1,5,2,6\right),\mathrm{cycle}\left(3,7,5,8\right)\right)\right)=\mathrm{quaternion_group}$
Signatures:
sts

 [Next: vierer_group] [Previous: dihedral_group] [Top]

## vierer_group

Description:

This symbol represents the Klein Vierer group of order 4, viewed as a permutation group of degree 4. It consists of the identity, (1,2)(3,4), (1,3)(2,4), and (1,4)(2,3).

Formal Mathematical property (FMP):
$\mathrm{group}\left(\mathrm{right_compose},\mathrm{permutation}\left(\mathrm{cycle}\left(1,2\right),\mathrm{cycle}\left(3,4\right)\right),\mathrm{permutation}\left(\mathrm{cycle}\left(1,3\right),\mathrm{cycle}\left(2,4\right)\right)\right)=\mathrm{vierer_group}$
Signatures:
sts

 [First: symmetric_group] [Previous: quaternion_group] [Top]