论文标题
对称组中的“功率”共轭方程
A "power" conjugate equation in the symmetric group
论文作者
论文摘要
首先，我们考虑一般“立方”方程的解决方案A_ {1} X^{r1} a_ {2} x^{r2} a_ {3} x^x^{r3} = 1 （在{1，-1}中使用R1，R2，R3）在对称组S_ {N}中。在某些情况下，该方程可以被重写为aya^{ - 1} = y^{2}或Aya^{ - 1} = y^{ - 2}，其中a in s in s_ {n}取决于a_ {i} s in s_ {i} s，s_ {n}中的新不知名y in s_ {n}中是x（n}中的一个poper的产品x（或x（或x）的产品， a_ {i}^{1}和a_ {i}^{ - 1}。使用组合参数和一些基本数字理论事实，我们获得了有关所谓的幂共轭方程的解决方案aya^{ - 1} = y^{e}的解决方案，其中e是整数指数。在某些条件下，解决方案正是在a的centralizer中的y^{e-1} = 1的解决方案。
First we consider the solutions of the general "cubic" equation a_{1}x^{r1}a_{2}x^{r2}a_{3}x^{r3}=1 (with r1,r2,r3 in {1,-1}) in the symmetric group S_{n}. In certain cases this equation can be rewritten as aya^{-1}=y^{2} or as aya^{-1}=y^{-2}, where a in S_{n} depends on the a_{i}'s and the new unknown permutation y in S_{n} is a product of x (or x^{-1}) and one of the permutations a_{i}^{1} and a_{i}^{-1}. Using combinatorial arguments and some basic number theoretical facts, we obtain results about the solutions of the so-called power conjugate equation aya^{-1}=y^{e} in S_{n}, where e is an integer exponent. Under certain conditions, the solutions are exactly the solutions of y^{e-1}=1 in the centralizer of a.
