Prime power Sylow numbers

Khalili Asboei, Alireza

doi:10.22103/jmmr.2026.26388.1909

Prime power Sylow numbers

Document Type : Research Paper

Author

Alireza Khalili Asboei

Department of Mathematics Education, Farhangian University, P.O. Box 14665-889, Tehran, Iran

10.22103/jmmr.2026.26388.1909

Abstract

In this paper, we study finite groups with a prime-power number of Sylow $p$-subgroups. Motivated by the work of Yang et al.~(2022), who characterized non-solvable groups with a prime-power number of Sylow $2$-subgroups, we investigate the corresponding problem for odd primes. We prove that if a finite group $G$ has a non-abelian composition factor whose order is divisible by an odd prime $p$, and the number of Sylow $p$-subgroups is a prime power, then $p$ must be a Mersenne prime and $n_p(G)=2^k$ for some integer $k\ge 2$.

Keywords

finite group

non-solvable group

Sylow number

Subjects

Algebra

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