Inner holomorph of a group: Difference between revisions

Revision as of 13:32, 24 September 2009

Definition

Let $G$ be a group. The inner holomorph of $G$ can be defined as the semidirect product $G ⋊ I n n (G)$ where $I n n (G)$ is the inner automorphism group with the usual action.

It is a subgroup of the holomorph $G ⋊ A u t (G)$ and is a quotient of the direct product $G \times G$ .

Facts

When $G$ is an abelian group, group of nilpotency class two, group whose center is a direct factor, or centerless group, this is isomorphic to the central product of two copies of $G$ with the center $Z (G)$ of both copies identified: $G *_{Z (G)} G$ . In other words, it is the quotient of $G \times G$ by the subgroup ${(g, g^{- 1}) ∣ g \in Z (G)}$ .

If $G$ is a group whose center is a direct factor, this group is isomorphic to the direct product of $G$ and $I n n (G)$ .

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 ==Definition==
-Let <math>G</math> be a [[group]]. The '''inner holomorph''' of <math>G</math> can be defined in the following equivalent ways:
+Let <math>G</math> be a [[group]]. The '''inner holomorph''' of <math>G</math> can be defined as the [[semidirect product]] <math>G \rtimes \operatorname{Inn}(G)</math> where <math>\operatorname{Inn}(G)</math> is the [[inner automorphism group]] with the usual action.
-# It is the [[semidirect product]] <math>G \rtimes \operatorname{Inn}(G)</math> where <math>\operatorname{Inn}(G)</math> is the [[inner automorphism group]] with the usual action.
+It is a subgroup of the [[holomorph of a group|holomorph]] <math>G \rtimes \operatorname{Aut}(G)</math> and is a quotient of the [[direct product]] <math>G \times G</math>.
-# It is the [[central product]] of two copies of <math>G</math> with the [[center]] <math>Z(G)</math> of both copies identified: <math>G *_{Z(G)} G</math>. In other words, it is the quotient of <math>G \times G</math> by the subgroup <math>\{ (g,g^{-1}) \mid g \in Z(G) \}</math>.
+==Facts==
+When <math>G</math> is an [[abelian group]], [[group of nilpotency class two]], [[group whose center is a direct factor]], or [[centerless group]], this is isomorphic to the [[central product]] of two copies of <math>G</math> with the [[center]] <math>Z(G)</math> of both copies identified: <math>G *_{Z(G)} G</math>. In other words, it is the quotient of <math>G \times G</math> by the subgroup <math>\{ (g,g^{-1}) \mid g \in Z(G) \}</math>.
-It is a subgroup of the [[holomorph of a group|holomorph]] <math>G \rtimes \operatorname{Aut}(G)</math>.
 If <math>G</math> is a [[group whose center is a direct factor]], this group is isomorphic to the [[direct product]] of <math>G</math> and <math>\operatorname{Inn}(G)</math>.