group3 http://www.openmath.org/cd http://www.openmath.org/cd/group3.ocd 2006-06-01 2004-06-01 1 2 experimental A CD of group constructions Written by Arjeh M. Cohen 2004-02-20. automorphism_group This is a function with a single argument which must be a group. It refers to the automorphism group of its argument. direct_product This is an n-ary function whose arguments must be groups. It refers to the direct product of its arguments. direct_power This is a binary function whose first argument should be a group G and whose second argument should be a natural number n. It refers to the direct product of n copies of G. sylow_subgroup This symbol represents a binary function with two arguments, the first is a group G and the second a prime number p. When applied to G and p, it represents a Sylow p-subgroup of G (which is unique up to conjugacy in G). derived_subgroup The unary function whose value is the subgroup of argument generated by all products of the form xyx^-1y^-1. d in the derived subgroup of G if and only if there exist lists x,y of elements of G of equal length such that d is the product x_1 y_1 x_1^(-1) y_1^(-1) ... x_n y_n x_n^(-1) y_n^(-1). -1 -1 1 quotient_group The binary function whose value is the factor group of the first argument by the second, assuming the second is normal in the first. center This symbols represents a unary function whose argument should be a group G. Its value is the biggest subgroup of G all of whose elements commute with all elements of G. d is in the center of G if and only if for all g in G we have g d= d g. centralizer This symbols represents a binary function whose first argument should be a group G and whose second argument should be an element g or a list of elements L of the group G. Its value is the subgroup of G of all elements commuting with g or, if the second argument is a list, all elements of L. d is in the centralizer of g in G if and only if g d= d g. free_group This symbol represents a unary function. The argument is a list or a set. When evaluated on such an argument, the function represents the free group generated by the entries of the list or set. The free group on the letters a, b: GL This symbol is a function with one argument, which should be a vector space or a module V. When applied to V it represents the group of all invertible linear transformations of V. SL This symbol is a function with one argument, which should be a a module V over a commutative ring. When applied to V it represents the group of all invertible linear transformations of V of determinant 1. GLn This symbol is a function with two arguments. The first should be a positive integer n, the second a field F. When applied to n and F it represents the group of all invertible linear transformations of the vector space over F of dimension n. SLn This symbol is a function with two arguments. The first should be a positive integer n, the second a field F. When applied to n and F it represents the group of all invertible linear transformations of the vector space over F of dimension n having determinant 1. normalizer This symbols represents a binary function whose first argument should be a group G and whose second argument should be a set of elements or a subgroup L of the group G. Its value is the subgroup of G of all elements normalizing L. d is in the normalizer of X in G if and only if g X= X g. symmetric_group This symbol is a function with one argument, which should be a set X. When applied to a set X it represents the group of all permutations on X . symmetric_groupn This symbol is a function with one argument, which should be a natural number n. When applied to n it represents the group of all permutations on the set {1,2,... ,n}. The carrier set of symmetric_groupn(k) consists of all permutations with support in the integers {1,...,k}. alternating_group This symbol is a function with one argument, which should be a set X. When applied to a set X it represents the group of all even permutations on X . alternatingn This symbol is a function with one argument, which should be a natural number n. When applied to n it represents the group of all even permutations on the set {1,2, ...,n}. invertibles This symbol is a function with one argument, which should be a monoid M. When applied to M it represents the group of all invertible elements of M.