The Quantities About Characters And The Structure Of A Finite Group

In this paper,we disscuss the relation of the quantities about characters and the structure of a finite group. In section 2,we define μ(G) = |G|/{Irr(G)| and investigate the effect of μ(G) to the structure of G with some conditions.We have the following results:Theorem 2.2 Let G be non-solvable group,then μ(G) ≥ 12 , and μ(G) = 12 if andonly if G (?) A₅×A,where A is abelian.Theorem 2.3 Let G be non-solvable group,then μ(G)≥120/7 , or G (?)A₅×Ai, orthere is N (?) G, such that G/N (?)SL(2,5). where A is abelian group.Theorem 2.4 Let G be non-abelian finite group. Then(l)If |G/G'| = 2,then μ(G) ≥ 2.And μ(G) = 2 if and only if G (?) S3.(2)If |G/G'| = 3,then μ{G) ≥ 3.And μ(G) = 3 if and only if G (?) A₄.Meanwhile, we study the structure and property of the group G when μ(G) are some small integers.For example:Theorem 2.5 Let G be non-abelian finite group ,then the following conditions are equivalent:Theorem 2.6 μ(G) = 3 if and only if G/Z{G) (?) A₄,D₁₈ or 3 = b³ = c² = 1.ab= ba,c^-1ac = a^-1,c^-1bc = b^-1>, and for any x,y ∈ G, [x,y](?)Z^*(G)...