Classifying complements for groups. Applications
Annales de l'Institut Fourier, Volume 65 (2015) no. 3, p. 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 : https://doi.org/10.5802/aif.2958
Classification:  20B05,  20B35,  20D06,  20D40
Keywords: Matched pairs, bicrossed products, the classification of finite groups
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     author = {Agore, Ana-Loredana and Militaru, Gigel},
     title = {Classifying complements for groups. Applications},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {3},
     year = {2015},
     pages = {1349-1365},
     doi = {10.5802/aif.2958},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_3_1349_0}
}
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/item/AIF_2015__65_3_1349_0/

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