Embeddings of finite groups in B n /Γ k (P n ) for k=2,3
[Plongements de groupes finis dans B n /Γ k (P n ) pour k=2,3]
Annales de l'Institut Fourier, Tome 70 (2020) no. 5, pp. 2005-2025.

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 ).

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.

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DOI : https://doi.org/10.5802/aif.3380
Classification : 20F36,  20F18,  20F14,  20B35,  16S34
Mots clés : Groupes de tresses, quotients, plongement de groupes finis
@article{AIF_2020__70_5_2005_0,
     author = {Lima Gon\c{c}alves, Daciberg and Guaschi, John and Ocampo, Oscar},
     title = {Embeddings of finite groups in $B_n/\Gamma _k(P_n)$ for $k=2,3$},
     journal = {Annales de l'Institut Fourier},
     pages = {2005--2025},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {70},
     number = {5},
     year = {2020},
     doi = {10.5802/aif.3380},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3380/}
}
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, Tome 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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