group4 http://www.openmath.org/cd http://www.openmath.org/cd/group4.ocd 2006-06-01 2004-06-01 1 2 experimental A CD of sets constructed from groups Written by Arjeh M. Cohen 2004-03-02. Edited AMC 2004-03-05 conjugacy_classes This symbol represents a unary function whose argument should be a group. Its value on a group is the set of conjugacy classes of that group. conjugacy_class_representatives This symbol represents a unary function whose argument should be a group. Its value on a group is a set of representatives of the conjugacy classes of that group. are_conjugate This symbol represents a boolean ternary function whose first argument is a group G and whose second and third arguments are elements x and y of G. Its value on G, x, and y is true if and only if x and y are conjugate in G. x and y are conjugate if and only if there is h in G such that x = h y h^(-1). -1 conjugacy_class This symbol represents a binary function, whose first argument is a group G and whose second argument is an element x of G. Its value on G and x is the set of elements which are conjugate to x in G. The conjugacy class in G of h is the subset {g^(-1) h g | g in G} of G. left_transversal The binary function whose value is a set of representatives for the left cosets of the second argument as a subgroup of the first. right_transversal The binary function whose value is a set of representatives for the right cosets of the second argument as a subgroup of the first. left_coset This symbol represents a ternary function whose first argument is a group G, whose second argument is a subgroup H of G, and whose third argument is an element x of G. Its value on G, H, and x is the left coset of H in G containing x, that is, the set x H. right_coset This symbol represents a ternary function whose first argument is a group G, whose second argument is a subgroup H of G, and whose third argument is an element x of G. Its value on G, H, and x is the right coset of H in G containing x, that is, the set H x. left_cosets The binary function whose value is the set of left cosets of the second argument in the first. right_cosets The binary function whose value is the set of right cosets of the second argument in the first. left_coset_representative This symbol represents a quaternary function whose first argument is a group G, whose second argument is a subgroup H of G, whose third argument is left_transversal T of H in G, and whose fourth argument is an element of G. It assigns to G, H, T, g the element of t of T representing the left coset of H containing g, that is, t H = g H . right_coset_representative This symbol represents a quaternary function whose first argument is a group G, whose second argument is a subgroup H of G, whose third argument is right_transversal T of H in G, and whose fourth argument is an element of G. It assigns to G, H, T, g the element of t of T representing the right coset of H containing g, that is, H t = H g.