Finite Simple Groups of Bounded Subgroup Chain Length

Alladi, Krishnaswami; Solomon, Ronald; Turull, Alexandre

doi:10.1006/jabr.2000.8371

Cited by 9 publications

(25 citation statements)

References 8 publications

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“…Example 2.13. By [1,Corollary 4.2], there are infinitely primes p such that Ω(p 2 −1) ≤ 21. For every such prime p, the exponent of PSL 2 (p) divides N p := p(p 2 − 1), so the word map (x, y) → x Np y Np cannot be surjective on PSL 2 (p) (its image consists only of the identity element); on the other hand, π(N p ) ≤ Ω(N p ) ≤ 22.…”

Section: Preliminariesmentioning

confidence: 99%

Surjective word maps and Burnside’s $$p^aq^b$$ p a q b theorem

et al. 2018

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We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map (x, y) → x N y N is surjective on every finite non-abelian simple group; if N is an odd integer, then the word map (x, y, z) → x N y N z N is surjective on every finite quasisimple group. These generalize classical theorems of Burnside and Feit-Thompson. We also prove asymptotic results about the surjectivity of the word map (x, y) → x N y N that depend on the number of prime factors of the integer N .

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Section: Preliminariesmentioning

confidence: 99%

Surjective word maps and Burnside’s $$p^aq^b$$ p a q b theorem

et al. 2018

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“…And are there infinitely many primes q such that l(PSL(2, q)) = 5 and 7 ≤ Ω(|PSL(2, q)|) ≤ 8? Through intense numbertheoretic analysis, Alladi et al [1] were able to prove that there exist infinitely many primes q such that l(PSL(2, q)) ≤ 20, but this result does not necessarily provide a positive answer to either question. In fact, the problem is deeper than one might suspect.…”

Section: Final Remarksmentioning

confidence: 99%

On the length of finite simple groups having chain difference one

Petrillo

2006

Arch. Math.

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A corollary to the known classification of finite simple groups having chain difference one (Brewster et al., J. Algebra 160, 179-191 (1993)) is that the length of such groups is either four or five. We prove a partial converse of the corollary to obtain a more precise description of these groups, the number of which depends on the unsolved prime k-tuples conjecture. (2000). 20E15. Mathematics Subject Classification

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“…Notice that the previous theorem cannot be deduced from Theorem 1. For example ρ(PSL(2, p)) = 2 for every prime p and it follows from [1,Corollary 4.2] that there are infinitely many primes p such that | PSL(2, p)| is divisible by at most 20 primes.…”

Section: Introductionmentioning

confidence: 99%