Sylow Numbers of Finite Groups

AbstractA natural number n is said to be a Sylow number for a finite group G if n is the of Sylow p-subgroups of G for some prime p. We initiate in this paper a systematic study on how arithmetical conditions on Sylow numbers influence the group structure. This new perspective leads us to prove, among others, the following two new results, which confirm a conjecture by Huppert and generalize some classical theorems in group theory. (1) A finite group G is p-nilpotent if and only if p is prime to every Sylow number of G. (2) If all Sylow numbers of a finite group G are square-free then G has at most one non-cyclic chief factor, and furthermore the possible non-cyclic chief factor is isomorphic to PSL(2,p) for some prime number p congruent to 5 modulo 8. Thus finite solvable groups with only square-free Sylow numbers are supersolvable. We will also pose some open problems