permgp2 http://www.openmath.org/cd http://www.openmath.org/cd/permgp2.ocd 2004-06-01 1 1 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 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}. The permutation group generated by (1,2) and (2,3) is equal to the symmetric group on {1,2,3}. 12 23 3 alternating_group 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}. The permutation group generated by (1,2,3) and (3,4,5) is equal to the alternating group on {1,2,3,4,5}. 123 345 5 cyclic_group 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). dihedral_group 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. 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. The dihedral group on 3 (letters) coincides with the symmetric group on 3 (letters). 3 3 quaternion_group 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.) 1324 5867 1526 3758 vierer_group 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). 12 34 13 24