Inner holomorph of a group: Difference between revisions

Latest revision as of 17:49, 24 September 2009

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 $G \times G$ .

When $G$ is an group having an automorphism whose restriction to the center is the inverse map, this is isomorphic to the central product of two copies of $G$ with the center $Z (G)$ of both copies identified: $G *_{Z (G)} G$ .

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

@@ Line 1: / Line 1: @@
 ==Definition==
-Let <math>G</math> be a [[group]]. The '''inner holomorph''' of <math>G</math> 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.
+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 a subgroup of the [[holomorph of a group|holomorph]] <math>G \rtimes \operatorname{Aut}(G)</math>.
+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>.
+==Facts==
+When <math>G</math> is an [[group having an automorphism whose restriction to the center is the inverse map]], 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>.
 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>.