Arithmetical conditions on invariant Sylow numbers
Let $G$ be a finite group and let $A$ be a subgroup of the automorphism group of $G$. Let $\pi(G)$ be the set of all prime divisors of $|G|$. For every $p\in \pi(G)$, define $ \nu_p^A(G)$ to be the number of Sylow $p$-subgroups of $G$ which are invariant under the action of $A$, and let $\mathrm{sn}_A(G)$ be the set of all values of $\nu_p^A(G)$ as $p$ runs through all primes in $\pi(G)$. The main result of the paper consists of results characterizing when $G$ is solvable under the condition that $(|A|,|G|)=1$ and $\mathrm{sn}_A(G)$ contains only prime powers, or only squarefree numbers, or both.
scientific article; zbMATH DE number 6879889
scientific article