一类共轭类长度集合的长度为3的有限群

A Type of Finite Groups With Three Conjugacy Lengths

证明了:若G为有限群,且|cd(G)|=|cs(G)|=3,则G=H×A.其中A是交换群,H是非交换p-群且|cs(H)|=3,或H=KL,K■H,(|K|,|L|)=1,K是非交换p-群且|cs(K)|=2,L是交换群,Z(K)=Z(H)∩K,H/Z(H)是Frobenius群,并且|cd(K)|=2,c(K)=2.

This paper is based on the classification of finite groups with three conjugacy lengths to find out which have three character degrees.If G is a finite group with three conjugacy lengths and three character degrees,then G=H×A,H is an unabelian p-group,A is abelion;or H=KL,K■H,(|K|,|L|)=1,K is an unabelian p-group with two conjugacy lengths and L is abelian,Z(K)=Z(H)∩K,H/Z(H)is a Frobenius group,and K have two different character degrees,the nilpotent class of K is two.