# Holomorph of a group

A concept in group theory which arose in connection with the following problem. Is it possible to include any given group $G$ as a normal subgroup in some other group so that all the automorphisms of $G$ are restrictions of inner automorphisms of this large group? To solve a problem of this kind, a new group $\Gamma$ is constructed using $G$ and its automorphism group $\Phi ( G)$. The elements of $\Gamma$ are pairs $( g, \phi )$ where $g \in G$, $\phi \in \Phi ( G)$, and composition of pairs is defined by the formula

$$( g _ {1} , \phi _ {1} ) ( g _ {2} , \phi _ {2} ) = \ ( g _ {1} g _ {2} ^ {\phi _ {1} ^ {-} 1 } ,\ \phi _ {1} \phi _ {2} ),$$

where $g _ {2} ^ {\phi _ {1} ^ {-} 1 }$ is the image of $g _ {2}$ under $\phi _ {1} ^ {-} 1$. The group $\Gamma$( or a group isomorphic to it) is called the holomorph of $G$. The set of pairs of the form $( g, \epsilon )$, where $\epsilon$ is the identity element of $\Phi ( G)$, constitutes a subgroup that is isomorphic to the original group $G$. In a similar manner, the pairs of the form $( e , \phi )$, where $e$ is the identity element of $G$, constitute a subgroup isomorphic to the group $\Phi ( G)$. The formula

$$( e, \phi ^ {-} 1 ) ( g, \epsilon ) ( e, \phi ) = \ ( g ^ \phi , \epsilon )$$

shows that $\Gamma$ is in fact a solution of the problem posed above.