Classifying complements for groups. Applications

Agore, Ana-Loredana; Militaru, Gigel

doi:10.5802/aif.2958

Classifying complements for groups. Applications
[Classification des compléments pour les groupes. Applications]

Agore, Ana-Loredana ^1,² ; Militaru, Gigel ³

¹ Faculty of Engineering, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels (Belgium)
² Permanent address: Department of Applied Mathematics, Bucharest University of Economic Studies, Piata Romana 6, RO-010374 Bucharest 1 (Romania)
³ Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, RO-010014 Bucharest 1 (Romania)

Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1349-1365.

Résumé
Abstract

Soit $G$ un groupe et $A \leq G$ un sous-groupe de $G$ . Un $A$ –complément de $G$ est un sous-groupe $H$ de $G$ tel que $G = A H$ et $A \cap H = {1}$ . Le problème auquel on s’intéresse est de classifier et décrire tous les $A$ –compléments de $G$ . Nous donnons la réponse à ce problème en trois étapes. Fixons $H$ un $A$ –complément de $G$ et soient $(▹, ◃)$ les actions canoniques associées à la factorisation $G = A H$ . On commence par déformer $H$ en un nouveau $A$ –complément $H_{r}$ à l’aide d’une certaine fonction $r : H \to A$ appelée fonction de déformation de $(A, H, ▹, ◃)$ . Ensuite on donne la description de tous les $A$ –compléments : $ℍ \leq G$ est un $A$ –complément de $G$ si et seulement si $ℍ$ est isomorphe à $H_{r}$ pour une certaine fonction de déformation $r : H \to A$ . Enfin, la classification des $A$ –compléments prouve qu’il existe une bijection entre les classes d’isomorphisme de tous les $A$ –compléments de $G$ et un objet cohomologique $𝒟 (H, A | (▹, ◃))$ . Comme application, on démontre que la formule qui calcule le nombre de classes d’isomorphisme des groupes d’ordre $n$ peut être retrouvée à partir de la factorisation $S_{n} = S_{n - 1} C_{n}$ .

Let $A \leq G$ be a subgroup of a group $G$ . An $A$ –complement of $G$ is a subgroup $H$ of $G$ such that $G = A H$ and $A \cap H = {1}$ . The classifying complements problem asks for the description and classification of all $A$ –complements of $G$ . We shall give the answer to this problem in three steps. Let $H$ be a given $A$ –complement of $G$ and $(▹, ◃)$ the canonical left/right actions associated to the factorization $G = A H$ . First, $H$ is deformed to a new $A$ –complement of $G$ , denoted by $H_{r}$ , using a deformation map $r : H \to A$ of the matched pair $(A, H, ▹, ◃)$ . Then the description of all complements is given: $ℍ$ is an $A$ –complement of $G$ if and only if $ℍ$ is isomorphic to $H_{r}$ , for some deformation map $r : H \to A$ . Finally, the classification of complements proves that there exists a bijection between the isomorphism classes of all $A$ –complements of $G$ and a cohomological object $𝒟 (H, A | (▹, ◃))$ . As an application we show that the theoretical formula for computing the number of isomorphism types of all groups of order $n$ arises only from the factorization $S_{n} = S_{n - 1} C_{n}$ .

DOI : 10.5802/aif.2958

Classification : 20B05, 20B35, 20D06, 20D40
Keywords: Matched pairs, bicrossed products, the classification of finite groups
Mot clés : Paires appariées, produits (bi)croisés, classification des groupes finis.

Affiliations des auteurs :

Agore, Ana-Loredana ^{1, 2} ; Militaru, Gigel ³

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     pages = {1349--1365},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Agore, Ana-Loredana; Militaru, Gigel. Classifying complements for groups. Applications. Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1349-1365. doi : 10.5802/aif.2958. https://aif.centre-mersenne.org/articles/10.5802/aif.2958/

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