GENERATION OF SECOND MAXIMAL SUBGROUPS AND THE EXISTENCE OF SPECIAL PRIMES
Forum of Mathematics, Sigma, Tome 5 (2017)
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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.
@article{10_1017_fms_2017_21,
author = {TIMOTHY C. BURNESS and MARTIN W. LIEBECK and ANER SHALEV},
title = {GENERATION {OF} {SECOND} {MAXIMAL} {SUBGROUPS} {AND} {THE} {EXISTENCE} {OF} {SPECIAL} {PRIMES}},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {5},
year = {2017},
doi = {10.1017/fms.2017.21},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2017.21/}
}
TY - JOUR AU - TIMOTHY C. BURNESS AU - MARTIN W. LIEBECK AU - ANER SHALEV TI - GENERATION OF SECOND MAXIMAL SUBGROUPS AND THE EXISTENCE OF SPECIAL PRIMES JO - Forum of Mathematics, Sigma PY - 2017 VL - 5 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.1017/fms.2017.21/ DO - 10.1017/fms.2017.21 LA - en ID - 10_1017_fms_2017_21 ER -
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TIMOTHY C. BURNESS; MARTIN W. LIEBECK; ANER SHALEV. GENERATION OF SECOND MAXIMAL SUBGROUPS AND THE EXISTENCE OF SPECIAL PRIMES. Forum of Mathematics, Sigma, Tome 5 (2017). doi: 10.1017/fms.2017.21
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