GENERATION OF SECOND MAXIMAL SUBGROUPS AND THE EXISTENCE OF SPECIAL PRIMES
Let $G$ be a finite almost simple group. It is well known that $G$ can be generated by three elements, a...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2017-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509417000214/type/journal_article |
| Summary: | 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. |
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| ISSN: | 2050-5094 |