20.86 (2022)

Suppose $G$ is a finite group and $p$ a prime such that the number $s_p(G) = 1+kp$ of Sylow $p$-subgroups $P$ of $G$ is greater than 1. By a theorem of Frobenius the set $G_p$ of all $p$-elements in $G$ has cardinality $|G_p| = |P| \cdot f_p(G)$ for some positive integer $f_p(G)$, and one easily gets that $f_p(G) = 1 + \ell(p-1) \leqslant k - \frac{k-1}{p-1}$. Usually, $\ell < k$ (but $\ell = k$ if $k < p$). Is always $\ell^p \geqslant k^{p-1}$?

Recently P. Gheri showed that $f_p(G)^p \geqslant s_p(G)^{p-1}$ if $G$ is $p$-solvable (Ann. Mat. Pura Appl. (4), 200 (2021), 1231–1243).