Classifying complements for groups. Applications
Annales de l'Institut Fourier, Volume 65 (2015) no. 3, pp. 1349-1365.

Let AG be a subgroup of a group G. An A–complement of G is a subgroup H of G such that G=AH and AH={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=AH. First, H is deformed to a new A–complement of G, denoted by H r , using a deformation map r:HA of the matched pair (A,H,,). Then the description of all complements is given: is an A–complement of G if and only if is isomorphic to H r , for some deformation map r:HA. 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 .

Soit G un groupe et AG un sous-groupe de G. Un A–complément de G est un sous-groupe H de G tel que G=AH et AH={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=AH. On commence par déformer H en un nouveau A–complément H r à l’aide d’une certaine fonction r:HA appelée fonction de déformation de (A,H,,). Ensuite on donne la description de tous les A–compléments : G est un A–complément de G si et seulement si est isomorphe à H r pour une certaine fonction de déformation r:HA. 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 .

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.
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)
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Agore, Ana-Loredana; Militaru, Gigel. Classifying complements for groups. Applications. Annales de l'Institut Fourier, Volume 65 (2015) no. 3, pp. 1349-1365. doi : 10.5802/aif.2958. http://www.numdam.org/articles/10.5802/aif.2958/

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