group5 http://www.openmath.org/cd/group5.ocd 2006-06-01 2004-07-07 1 1 experimental A CD of functions for relating group elements to their images in quotients. Written by Arjeh M. Cohen 2004-07-07 right_quotient_map This symbol is a binary function whose first argument is a group G and whose second argument is an subgroup H of G. When applied to G and H, its value is the natural quotient map from G to the quotient group G/H, sending x to the left coset xH of G. The image of an element x is the left coset of x with respect to H. left_quotient_map This symbol is a binary function whose first argument is a group G and whose second argument is an subgroup H of G. When applied to G and H, its value is the natural quotient map from G to the quotient group G/H, sending x to the right coset Hx of G. The image of an element x is the right coset of x with respect to H. The left and right quotients have a natural group structure if and only if H is a normal subgroup of G. homomorphism_by_generators This is a function with three arguments the first two of which must be groups F and K. The third argument should be a set or a list L of ordered pairs (lists of length 2). Each pair [x,y] from L consists of an element x from F and an element y from K. When applied to F, K, and L, the symbol represents the group homomorphism from F to K that maps the first entry x of each pair [x,y] to the second entry y of the same pair.