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.

References

[a1]	M. Hall jr., "The theory of groups" , Macmillan (1959)
[a2]	A.G. Kurosh, "Theory of groups" , 1 , Chelsea (1955) (Translated from Russian)