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 as a normal subgroup in some other group so that all the automorphisms of
are restrictions of inner automorphisms of this large group? To solve a problem of this kind, a new group
is constructed using
and its automorphism group
. The elements of
are pairs
where
,
, and composition of pairs is defined by the formula
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where is the image of
under
. The group
(or a group isomorphic to it) is called the holomorph of
. The set of pairs of the form
, where
is the identity element of
, constitutes a subgroup that is isomorphic to the original group
. In a similar manner, the pairs of the form
, where
is the identity element of
, constitute a subgroup isomorphic to the group
. The formula
![]() |
shows that is in fact a solution of the problem posed above.
Comments
References
[a1] | M. Hall jr., "The theory of groups" , Macmillan (1959) |
[a2] | A.G. Kurosh, "Theory of groups" , 1 , Chelsea (1955) (Translated from Russian) |
Holomorph of a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Holomorph_of_a_group&oldid=12406