Classifying complements for groups. Applications

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 $A\cap H=\{1\}$. The \emph{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 $G=AH$. To start with, $H$ is deformed to a new $A$-complement of $G$, denoted by $H_r$, using a certain map $r:H\to A$ called a deformation map of the matched pair $(A, H, \triangleright, \triangleleft)$\mathrm{.}Thenthedescriptionofallcomplementsisgiven:${\mathbb H}$isan$A$-complementof$G$ifandonlyif${\mathbb H}$isisomorphicto$H_{r}$,forsomedeformationmap$r: H \to A$. 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 ${\mathcal D} , (H, A , | ,(\triangleright, \triangleleft))$. 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 $S_n=S_{n-1}{C}_n$.