Mechanization Of Mathematics Of Operations And Proof On Permutation Groups

The algorithm study of group theory is a meaningful problem. Most of the groups which we use in practice are so complex that we have to realize their operations by computer. This paper realizes the operations and proof on permutation groups by computer algebra system—Mathematica.We design different algorithms and realize them by Mathematica for different problems, such as some basic operations on permutation groups, the operations and generation of subgroups and the operations of a group on a set.Classify the elements of A_n by conjugation and compute all the possible sums of these classes' orders except the order of the identity element's conjugation class. Add 1 to each sum, and then divide |A_n| by the results. If there is a number k among the results which can divide |A_n |, only the groups generated by those conjugation classes whose orders' sum is k may be the untrivial normal subgroup of A_n. Based on this theory, we present an algorithm to judge whether the alternating group A_n with given n is a simple group or not with computer algebra method. The results that A_n (n ≠ 4) is a simple group can be got quickly when n is less than 10.Caley theory presents the relationship between an abstract group G and a specific group S_n. We can get all groups with order n if we get all subgroups with order n of group S_n. This paper designs an algorithm to find all subgroups of a symmetric group and to complete their conjugation classification, as examples, we complete the conjugation classification of all subgroups of S_n(n ≤ 7).