permgp2
http://www.openmath.org/cd
http://www.openmath.org/cd/permgp2.ocd
2004-06-01
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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}.
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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}.
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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).
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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.)
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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).
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