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@article{Liebeck:2017:10.1017/fms.2017.21,
author = {Liebeck, MW and Burness, TC and Shalev, A},
doi = {10.1017/fms.2017.21},
journal = {Forum of Mathematics, Sigma},
title = {Generation of second maximal subgroups and the existence of special primes},
url = {http://dx.doi.org/10.1017/fms.2017.21},
volume = {5},
year = {2017}
}

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TY  - JOUR
AB  - 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. Inthis paper, we consider subgroups at the next level of the subgroup lattice—the so-called secondmaximal subgroups. We prove that with the possible exception of some families of rank 1 groupsof Lie type, the number of generators of every second maximal subgroup of G is bounded by anabsolute constant. We also show that such a bound holds without any exceptions if and only if thereare only finitely many primes r for which there is a prime power q such that (qr − 1)/(q − 1)is prime. The latter statement is a formidable open problem in Number Theory. Applications torandom generation and polynomial growth are also given.
AU  - Liebeck,MW
AU  - Burness,TC
AU  - Shalev,A
DO  - 10.1017/fms.2017.21
PY  - 2017///
SN  - 2050-5094
TI  - Generation of second maximal subgroups and the existence of special primes
T2  - Forum of Mathematics, Sigma
UR  - http://dx.doi.org/10.1017/fms.2017.21
UR  - http://hdl.handle.net/10044/1/50423
VL  - 5
ER  -

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