Holomorph of a group

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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 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

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.



[a1] M. Hall jr., "The theory of groups" , Macmillan (1959)
[a2] A.G. Kurosh, "Theory of groups" , 1 , Chelsea (1955) (Translated from Russian)
How to Cite This Entry:
Holomorph of a group. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.N. Remeslennikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article