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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|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.

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Hall jr.,   "The theory of groups" , Macmillan  (1959)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.G. Kurosh,   "Theory of groups" , '''1''' , Chelsea  (1955)  (Translated from Russian)</TD></TR></table>