Sylow Numbers of Finite Groups

✍ Scribed by J.P. Zhang

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 661 KB
• Volume: 176
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1995.1235

✦ Synopsis

A natural number (n) is said to be a Sylow number for a finite group (G) if (n) is the number 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 (\operatorname{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.x (c) 1995 Academic Press. Inc.