OpenMath Content Dictionary: permgp2

Canonical URL:
http://www.openmath.org/cd/permgrp.ocd
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
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}.
  
group ⁡ ( right_compose , permutation ⁡ ( cycle ⁡ ( 1 , 2 ) ) , permutation ⁡ ( cycle ⁡ ( 2 , 3 ) ) ) = symmetric_group ⁡ ( 3 )
Signatures:
sts

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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}.
  
group ⁡ ( right_compose , permutation ⁡ ( cycle ⁡ ( 1 , 2 , 3 ) ) , permutation ⁡ ( cycle ⁡ ( 3 , 4 , 5 ) ) ) = alternating_group ⁡ ( 5 )
Signatures:
sts

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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

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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).
  
dihedral_group ⁡ ( 3 ) = symmetric_group ⁡ ( 3 )
Signatures:
sts

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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):
  
group ⁡ ( right_compose , permutation ⁡ ( cycle ⁡ ( 1 , 3 , 2 , 4 ) , cycle ⁡ ( 5 , 8 , 6 , 7 ) ) , permutation ⁡ ( cycle ⁡ ( 1 , 5 , 2 , 6 ) , cycle ⁡ ( 3 , 7 , 5 , 8 ) ) ) = quaternion_group
Signatures:
sts

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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):
  
group ⁡ ( right_compose , permutation ⁡ ( cycle ⁡ ( 1 , 2 ) , cycle ⁡ ( 3 , 4 ) ) , permutation ⁡ ( cycle ⁡ ( 1 , 3 ) , cycle ⁡ ( 2 , 4 ) ) ) = vierer_group
Signatures:
sts

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