设G是有限群,ast(G)=Σp∈ T(G)vp(G)/|T(G)|(p是素数,vP(G)表示G的Sylowp-子群的个数,T(G)={p|vp(G)＞1,p=2,3,5}),本文证明了若 ast(G)＜ast(A5),则 G 可解;以及 G(≌)A5 当且仅当 ast(G)=ast(A5)且h(G)=h(A5),其中h(G)表示G中元的最高阶.
Average number of Sylow subgroups and solvability in a finite groups
Let G be a finite group.For a prime divisor p of G let vp(G)denote the number of Sylow p-subgroup of G.We set ast(G)=Σp∈ T(G)vp(G)/|T(G)|,where T(G)={p|vp(G)＞1,p=2,3,5}.In this paper,we prove that if ast(G)＜ast(A5),then G is solvable;and G(≌)A5 if and only if ast(G)=ast(A5),h(G)=h(A5),where h(G)=max{o(g)|g ∈ G}.
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