Prime power Sylow numbers

In this paper, we study finite groups with a prime-power number of Sylow \(p\)-subgroups. Motivated by the work of Yang et al.~(۲۰۲۲), who characterized non-solvable groups with a prime-power number of Sylow ۲-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 \(n_p(G)=۲^k\) for some integer \(k\ge ۲\).

کلیدواژه ها:

Finite group ، non-solvable group ، Sylow number