Algorithms for symmetric groups of simplexes

An algorithm is developed to solve a special case of reverse problems of linear systems, and as its application, a method is presented to solve optimal problem over the special orthogonal group. An algorithm is designed to find out an orthogonal matrix A such that A transforms a set of orthogonal vectors to another, and as its application, a method is given to generate subgroups of the orthogonal group preserving several orthogonal vectors fixed. It is prove that the symmetric (or rotational) group of the simplex Sn preserving its center fixed is isomorphic to the group consisting of all n×n permutational matrices. All algorithms in this paper can be realized by algebraic system CoCoA 4.3.