Finite Solvable Groups And Prime Divisor Set Of Conjugacy Class Lengths

The main problem discussed in this paper is the prime divisor set of the conjugacy class lengths and the structure of finite solvable groups. The relationship between ρ*(G)and σ*(G) of different groups and conditions have been discussed in this paper. The following conclusions have been proved:If|Δ(G)|≤3,then|ρ*(G)|≤3σ*(G);If Gis finite solvable group,n is the number of prime divisors p with G having abelian Sylow p-subgroup, m is the number of prime divisors p with G having non-abelian Sylow subgroups,if n≥2m-3,then|ρ*(G)|≤3σ*(G):If G is metabelian groups, supersolvable groups and metanilpotent groups,|ρ*(G)|≤3σ*(G);The following conclusion have been verified by the GAP software:for the groups of order at most2000,|ρ*(G)|≤2σ*(G):For the groups of at most2000except768,1024,1152,1280,1536,1792,1920,the number of groups satisfied|ρ*(G)|=2σ*(G) is5286. For the group Gn satisfied,the order n∈(2000,∝).