关于方程X^n=1在有限群内的解

On Solution of Equation Xn= 1 in Finite Groups

1903年Frobenius证明了:一个有限群G,如果n整除它的阶,则方程x^n=1在G内解的个数是n的倍数。与这个定理有关的一个猜想是:如果n整除有限群G的阶,且方程x^n=1在G内恰好有n个解,则这些解作成G的正规子群。关于这个猜想目前已有的两个结论如下:

In this paper, we discuss the conjecture that if the order of finite group G is divisible by integer n, and the equation Xa=1 has exactly n solutions in G> then tne solutions make up a normal subgroup of G. The main results are the following theorems. Theoreml. Let G a finite group, n= 2pa,p a prime, a>0. If the order of G is divisible by n, and the equation xn= 1 has exactly n solution in G, then the solutious make up a normal subgroup of G. Theorem Let G a finite group, n = paq, p andq are primes, a>0, q>pa. If the order of G is divisible by n, and the equation xa= 1 has exactly n solutions in G, then the solutions make up a normal subgroup of G. Theorem 3 Let G a finite group, n= 2paq,p and q are odd primes, a>0, q>pa. If the order of is divisible by n, and the equation xn= 1 has exactly n solutions in G, then the solutions make up a normal subgroup of G. Theorem 4. Let G a finite group of order g, and let 4tg or = 4 m , where m is an odd number and 3m, n is ang fortor of g .If the equation xn = l has exactly n solutions in G, them the solutions make up a normal subgroup of G.