Primitive Prime Divisors of Orders of Suzuki–Ree Groups

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 2ⁱ − 1 for any 1 ≤ i < 4m. It is easy to see that r divides 2^2m + 1, and so it divides one of the numbers a and b. It is proved that for every m > 5, each of a, b is divisible by some primitive prime divisor of 2^4m − 1. Similar results are obtained 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.