| Autor: |
TIMOTHY C. BURNESS, MARTIN W. LIEBECK, ANER SHALEV |
| Jazyk: |
angličtina |
| Rok vydání: |
2017 |
| Předmět: |
|
| Zdroj: |
Forum of Mathematics, Sigma, Vol 5 (2017) |
| Druh dokumentu: |
article |
| ISSN: |
2050-5094 |
| DOI: |
10.1017/fms.2017.21 |
| Popis: |
Let $G$ be a finite almost simple group. It is well known that $G$ can be generated by three elements, and in previous work we showed that 6 generators suffice for all maximal subgroups of $G$ . In this paper, we consider subgroups at the next level of the subgroup lattice—the so-called second maximal subgroups. We prove that with the possible exception of some families of rank 1 groups of Lie type, the number of generators of every second maximal subgroup of $G$ is bounded by an absolute constant. We also show that such a bound holds without any exceptions if and only if there are only finitely many primes $r$ for which there is a prime power $q$ such that $(q^{r}-1)/(q-1)$ is prime. The latter statement is a formidable open problem in Number Theory. Applications to random generation and polynomial growth are also given. |
| Databáze: |
Directory of Open Access Journals |
| Externí odkaz: |
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