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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2504.04729 |
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| _version_ | 1866917979034222592 |
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| author | Grechkoseeva, Maria |
| author_facet | Grechkoseeva, Maria |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_04729 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On primitive prime divisors of the orders of Suzuki-Ree groups (corrected version) Grechkoseeva, Maria Group Theory 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. |
| title | On primitive prime divisors of the orders of Suzuki-Ree groups (corrected version) |
| topic | Group Theory |
| url | https://arxiv.org/abs/2504.04729 |