New bounds for numbers of primes in element orders of finite groups

TL;DR

This paper establishes new upper bounds on the number of distinct prime divisors in element orders of finite solvable groups, improving existing bounds and providing insights into the structure of such groups.

Contribution

The paper introduces a linear bound of 5n on the number of primes in element orders of finite solvable groups, refining previous results and extending to arbitrary finite groups.

Findings

Proved that $ ho(n) \\leq 5n$ for all $n \\geq 1$.

Improved bounds on prime divisors in element orders of finite groups.

Enhanced understanding of the prime structure in finite solvable groups.

Abstract

Let $\rho(n)$ denote the maximal number of different primes that may occur in the order of a finite solvable group $G$, all elements of which have orders divisible by at most $n$ distinct primes. We show that $\rho(n)\leq 5n$ for all $n\geq 1$. As an application, we improve on a recent bound by Hung and Yang for arbitrary finite groups.