Enrico Jabara

On Huppert's $\rho$-$\sigma$ conjecture for primes in the orders of elements

Let $G$ be a finite group, let $\rho(G)$ denote the number of distinct prime divisors of $|G|$, and let $\sigma(G)$ be the maximal number of distinct prime divisors of the order of an element of $G$. Let $\rho(n)=\max\{\rho(G) \mid G \text{ solvable}, \sigma(G)=n\}$. We prove that $\rho(n)\le 4n-1+\pi(2n)$ for all $n$,...