Probabilistic and Asymptotic Aspects of Finite Simple Groups

Liebeck, Martin W.

doi:10.1007/978-1-4471-4814-2_1

Cited by 7 publications

(5 citation statements)

References 71 publications

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“…Following the determination of the Möbius function for the simple small Ree groups we use it to determine d Γ (G) for a number of finitely presented groups and use these to prove a number of related results in probabilistic generation of the small Ree groups. Many of these results were previously known, see [17] for a recent survey in this area, an exception is the following which the author is unaware of appearing elsewhere in the literature.…”

Section: Isomorphismmentioning

confidence: 82%

The Möbius function of the small Ree groups

Pierro

2014

Preprint

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The Möbius function for a group, G, was introduced in 1936 by Hall in order to count ordered generating sets of G. In this paper we determine the Möbius function of the simple small Ree groups, R(q) = 2 G 2 (q) where q = 3 2m+1 for m > 0, using their 2-transitive permutation representation of degree q 3 + 1 and describe their maximal subgroups in terms of this representation. We then use this to determine |Epi(Γ, G)| for various Γ, such as F 2 or the modular group P SL 2 (Z), with applications to Grothendieck's theory of dessins d'enfants as well as probabilistic generation of the small Ree groups.

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Section: Isomorphismmentioning

confidence: 82%

The Möbius function of the small Ree groups

Pierro

2014

Preprint

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“…For a given 𝐵 ∈ , recall that there are |𝐴 × |∕|𝑁 𝐴 × (𝐵 × )| conjugates of 𝐵 in 𝐴. Combining (11) with Lemma 16 gives us…”

Section: Lemma 16mentioning

confidence: 98%

“…In the past few decades, there has been extensive research on random generation of finite and profinite groups with emphasis on finite simple groups. See, for instance, the survey articles [11,23] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Random generation of associative algebras

Sercombe,

Shalev

2023

Journal of London Math Soc

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There has been considerable interest in recent decades in questions of random generation of finite and profinite groups and finite simple groups in particular. In this paper, we study similar notions for finite and profinite associative algebras. Let be a finite field. Let be a finite‐dimensional, associative, unital algebra over . Let be the probability that two elements of chosen (uniformly and independently) at random will generate as a unital ‐algebra. It is known that if is simple, then as . We extend this result to a large class of finite associative algebras. For simple, we find the optimal lower bound for and we estimate the growth rate of in terms of the minimal index of any proper subalgebra of . We also study the random generation of simple algebras by two elements that have a given characteristic polynomial (resp. a given rank). In addition, we bound above and below the minimal number of generators of general finite algebras. Finally, we let be a profinite algebra over . We show that is positively finitely generated if and only if has polynomial maximal subalgebra growth. Related quantitative results are also established.

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“…Let B be the standard set of representatives of the conjugacy classes of maximal subalgebras of A. For a given B ∈ B, recall that there are (11) with Lemma 17 gives us…”

Section: Lemma 15mentioning

confidence: 99%

Random generation of associative algebras

Sercombe¹,

Shalev²

2020

Preprint

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There has been considerable interest in recent decades in questions of random generation of finite and profinite groups, and finite simple groups in particular. In this paper we study similar notions for finite and profinite associative algebras. Let k = Fq be a finite field. Let A be a finite dimensional, associative, unital algebra over k. Let P (A) be the probability that two elements of A chosen (uniformly and independently) at random will generate A as a unital k-algebra. It is known that, if A is simple, then P (A) → 1 as |A| → ∞. We prove an analogue of this result for A an arbitrary finite associative algebra. For A simple, we find the lower bound for P (A) and we estimate the growth rate of P (A) in terms of the minimal index m(A) of any proper subalgebra of A. We also study the random generation of A by two elements that have a given characteristic polynomial (resp. a given rank). In addition, we bound above and below the minimal number of generators of A. Finally, we let A be a profinite algebra over k. We show that A is positively finitely generated if and only if A has polynomial maximal subalgebra growth. Related quantitative results are also established.

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Probabilistic and Asymptotic Aspects of Finite Simple Groups

Cited by 7 publications

References 71 publications

The Möbius function of the small Ree groups

The Möbius function of the small Ree groups

Random generation of associative algebras

Random generation of associative algebras

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