The Conjugacy Classes And Order Of Generator Polynomial Of Rubik's Cube Group

In this paper,we mainly study two problems on Rubik's cube group.One is the conjugacy classes of Rubik's cube group,and the other is the order of generator polynomial for Rubik's cube group.The Rubik's cube group is a group generated by the six basic moves U,D,L,R,F and B with highly abstract.For the sake of convenience,we first express the generators of Rubik's cube group as permutations,and therefore the Rubik's cube group is isomorphic to a subgroup of48 S.Based on these results we can obtain more properties concerning the Rubik's cube group.This article is organized as follows:In Chapter 1,we elaborate the background and research status of this problem,and enumerate the main results in this article.In Chapter 2,we introduce some preliminaries which are needed throughout this paper,such as permutation,conjugacy(class),cycle type and some conclusions in permutation group.In Chapter 3,we label the48 S facelets with numbers 1,2,…,48 in the home position of Rubik's cube,and express the six basic moves as permutations,which shows that the Rubik's cube group is isomorphic to a subgroup of48 S.Then we establish three precise relations among the six basic moves U,D,L,R,F and B by the aid of multiset and the known conclusions on group.Finally,we classify all the(square)cross moves and the set of products of two arbitrary(square)cross moves.More precisely,we prove that all the(square)cross moves are conjugate each other in48 S,and the set of products of two arbitrary cross moves has only twelve conjugacy classes in48 S.Also we show that the set of products of two arbitrary left(right)square cross moves has only twelve conjugacy classes in48 S.In Chapter 4,we find out the order of generator polynomial for the Rubik's cube group using “The first law of cubology”.Consider as the number of elements in the Rubik's cube group,we give a new representation for the set containing all the orders of the Rubik's cube group.At last we prove that the order of generator polynomial for the Rubik's cube group is 1260 in theory.