Let and . We study the embedding of a given finite group in the quotient , where is the th Artin braid group, is the lower central series of the th pure braid group , and denotes the order of . If such an embedding exists, it is known that . In this paper, we show that if is a finite group for which , then embeds into . If , the result was proved independently by Beck and Marin. If , where the action is injective, is an odd prime, if , and divides and satisfies , we show that embeds into . If , this is a special case of another result of Beck and Marin. We also construct explicit embeddings in of the two non-Abelian groups of order .
Soient et . Nous étudions le plongement d’un groupe fini donné dans le quotient , où est le groupe de tresses d’Artin, est la série centrale descendante du groupe de tresses pures , et désigne l’ordre de . Si un tel plongement existe, on sait que pgcd. Si est un groupe fini pour lequel pgcd, nous montrons dans cet article que se plonge dans . Si , ce résultat a été démontré indépendamment par Beck et Marin. Si , où l’action est injective, est un nombre premier impair, si , et divise et vérifie pgcd, nous montrons que se plonge dans . Si , 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 dans .
Revised:
Accepted:
Published online:
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
@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/} }
TY - JOUR AU - Lima Gonçalves, Daciberg AU - Guaschi, John AU - Ocampo, Oscar TI - Embeddings of finite groups in $B_n/\Gamma _k(P_n)$ for $k=2,3$ JO - Annales de l'Institut Fourier PY - 2020 SP - 2005 EP - 2025 VL - 70 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3380/ DO - 10.5802/aif.3380 LA - en ID - AIF_2020__70_5_2005_0 ER -
%0 Journal Article %A Lima Gonçalves, Daciberg %A Guaschi, John %A Ocampo, Oscar %T Embeddings of finite groups in $B_n/\Gamma _k(P_n)$ for $k=2,3$ %J Annales de l'Institut Fourier %D 2020 %P 2005-2025 %V 70 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3380/ %R 10.5802/aif.3380 %G en %F AIF_2020__70_5_2005_0
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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