Embeddings of finite groups in B n /Γ k (P n ) for k=2,3
Annales de l'Institut Fourier, Volume 70 (2020) no. 5, pp. 2005-2025.

Let n3 and k2,3. We study the embedding of a given finite group G in the quotient B n /Γ k (P n ), where B n is the nth Artin braid group, Γ l (P n ) l is the lower central series of the nth pure braid group P n , and G denotes the order of G. If such an embedding exists, it is known that gcd(G,k!)=1. In this paper, we show that if G is a finite group for which gcd(G,k!)=1, then G embeds into BG /Γ k (PG ). If k=2, the result was proved independently by Beck and Marin. If G= p r θ d , where the action θ is injective, p is an odd prime, p5 if k=3, and d divides p-1 and satisfies gcd(d,k!)=1, we show that G embeds into B p r /Γ k (P p r ). If k=2, this is a special case of another result of Beck and Marin. We also construct explicit embeddings in B 9 /Γ 2 (P 9 ) of the two non-Abelian groups of order 27.

Soient n3 et k2,3. Nous étudions le plongement d’un groupe fini donné G dans le quotient B n /Γ k (P n ), où B n est le n e  groupe de tresses d’Artin, Γ l (P n ) l est la série centrale descendante du n e  groupe de tresses pures P n , et G désigne l’ordre de G. Si un tel plongement existe, on sait que pgcdG,k!=1. Si G est un groupe fini pour lequel pgcdG,k!=1, nous montrons dans cet article que G se plonge dans BG /Γ k (PG ). Si k=2, ce résultat a été démontré indépendamment par Beck et Marin. Si G= p r θ d , où l’action θ est injective, p est un nombre premier impair, p5 si k=3, et d divise p-1 et vérifie pgcdd,k!=1, nous montrons que G se plonge dans B p r /Γ k (P p r ). Si k=2, c’est un cas particulier d’un autre résultat de Beck et Marin. Nous construisons également des plongements explicites des deux groupes non-abéliens d’ordre 27 dans B 9 /Γ 2 (P 9 ).

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DOI: 10.5802/aif.3380
Classification: 20F36, 20F18, 20F14, 20B35, 16S34
Keywords: Braid groups, quotients, embedding of finite groups
Mot clés : Groupes de tresses, quotients, plongement de groupes finis

Lima Gonçalves, Daciberg 1; Guaschi, John 2; Ocampo, Oscar 3

1 Departamento de Matemática – IME-USP Rua do Matão 1010 CEP: 05508-090 – São Paulo – SP (Brazil)
2 Normandie Univ., UNICAEN, CNRS, LMNO 14000 Caen (France)
3 Universidade Federal da Bahia Departamento de Matemática – IME Av. Adhemar de Barros S/N CEP: 40170-110 – Salvador – BA (Brazil)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Lima Gonçalves, Daciberg; Guaschi, John; Ocampo, Oscar. Embeddings of finite groups in $B_n/\Gamma _k(P_n)$ for $k=2,3$. Annales de l'Institut Fourier, Volume 70 (2020) no. 5, pp. 2005-2025. doi : 10.5802/aif.3380. https://aif.centre-mersenne.org/articles/10.5802/aif.3380/

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