On primitive prime divisors of the orders of Suzuki-Ree groups (corrected version)

Abstract

There is a well-known factorization of the number $2^{2m}+1$, with $m$ odd, related to the orders of tori of simple Suzuki groups: $2^{2m}+1$ is a product of $a=2^m+2^{(m+1)/2}+1$ and $b=2^m-2^{(m+1)/2}+1$. By the Bang-Zsigmondy theorem, there is a primitive prime divisor of $2^{4m}-1$, that is, a prime $r$ that divides $2^{4m}-1$ and does not divide $2^i-1$ for any $i<4m$. It is easy to see that $r$ divides $2^{2m}+1$, and so it divides one of the numbers $a$ and $b$. The main objective of this paper is to show that for every $m>5$, each of $a$ and $b$ is divisible by some primitive prime divisor of $2^{4m}-1$. Also we prove similar results for primitive prime divisors related to the simple Ree groups. As an application, we find the independence and 2-independence numbers of the prime graphs of almost simple Suzuki-Ree groups.

Theorems & Definitions (10)

• Theorem 1: Bang--Zsigmondy
• Theorem 2
• Theorem 3
• Lemma 1
• proof
• Lemma 2
• proof
• Remark 1
• Remark 2
• Remark 3