Primitive prime divisors of orders of Suzuki–Ree groups

M. A. Grechkoseeva

Algebra i logika, Tome 62 (2023) no. 1, pp. 59-70

Résumé

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 $1\leqslant i4m$. 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.

Détail

Keywords: primitive prime divisor, Suzuki–Ree groups, prime graph.