Classifying complements for groups. Applications

Agore, Ana-Loredana; Militaru, Gigel

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

Soit $G$ un groupe et $A\le G$ un sous-groupe de $G$. Un $A$–complément de $G$ est un sous-groupe $H$ de $G$ tel que $G=AH$ et $A\cap H=\left\{1\right\}$. 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 $r:H\to A$ appelée fonction de déformation de $(A,H,▹,◃)$. Ensuite on donne la description de tous les $A$–compléments : $ℍ\le G$ 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 $A\le G$ be a subgroup of a group $G$. An $A$–complement of $G$ is a subgroup $H$ of $G$ such that $G=AH$ and $A\cap H=\left\{1\right\}$. 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 $r:H\to A$ 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 $S_n=S_{n-1}{C}_n$.

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