Normal-extensible not implies normal: Difference between revisions

Revision as of 21:18, 30 May 2009

This article gives the statement and possibly, proof, of a non-implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., normal-extensible automorphism) need not satisfy the second automorphism property (i.e., normal automorphism)
View a complete list of automorphism property non-implications | View a complete list of automorphism property implications
Get more facts about normal-extensible automorphism|Get more facts about normal automorphism

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal subgroup) need not satisfy the second subgroup property (i.e., normal-extensible automorphism-invariant subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about normal subgroup|Get more facts about normal-extensible automorphism-invariant subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal subgroup but not normal-extensible automorphism-invariant subgroup|View examples of subgroups satisfying property normal subgroup and normal-extensible automorphism-invariant subgroup

Statement

In terms of automorphism properties

A normal-extensible automorphism of a group (i.e., an automorphism that can always be extended for any embedding of the group as a normal subgroup of a bigger group) need not be a normal automorphism, i.e., it need not send every normal subgroup to itself.

In terms of subgroup properties

A normal subgroup of a group need not be a normal-extensible automorphism-invariant subgroup: i.e., there may be normal-extensible automorphisms of the group that do not leave the normal subgroup invariant.

Statement with symbols

We can have a group $G$ and a normal-extensible automorphism $σ$ of $G$ that is not a normal automorphism: in other words, there exists a normal subgroup $N$ of $G$ such that $σ (N) \neq N$ .

Related facts

Stronger facts

Normal not implies normal-extensible automorphism-invariant in finite: This is the stronger version, and the example outlined here in fact shows the stronger version.

Applications

Facts used

Proof

Example of the dihedral group

Further information: dihedral group:D8, subgroup structure of dihedral group:D8

Let $G$ be the dihedral group of order eight. Then, every automorphism of $G$ fixes every element of the center of $G$ , and also, the inner automorphism group of $G$ is maximal in the automorphism group of $G$ . Thus, by fact (1), every automorphism of $G$ is normal-extensible.

However, there is an automorphism of $G$ that interchanges the two normal Klein four-subgroups. Thus, these two normal subgroups are not invariant under this automorphism, and hence, we have an automorphism of $G$ that is normal-extensible but not normal.

Equivalently, the Klein four-subgroups are examples of normal subgroups that are not normal-extensible automorphism-invariant.

Example involving a simple complete group

Let $S$ be a simple complete group. In other words, $S$ is a centerless simple group such that every automorphism of $S$ is inner. Let $G = S \times S$ . By fact (2), the automorphism group of $G$ is the wreath product of $S$ with the symmetric group of degree two, which has $G$ , the inner automorphism group, as a subgroup of index two. Moreover, $G$ is centerless. Thus, by fact (1), we get that every automorphism of $G$ is normal-extensible.

However, the coordinate exchange automorphism of $G$ , that interchanges the two copies of $S$ , is not a normal automorphism because it interchanges these two normal subgroups. Thus, we have an example of a normal-extensible automorphism that is not normal.

Equivalently, either of the direct factors is an example of a normal subgroup that is not normal-extensible automorphism-invariant.

@@ Line 39: / Line 39: @@
 # [[uses::Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible]]
+# [[uses::Automorphism group of direct power of simple non-abelian group equals wreath product of automorphism group and symmetric group]]
 ==Proof==
+===Example of the dihedral group===
 {{further|[[Particular example::dihedral group:D8]], [[subgroup structure of dihedral group:D8]]}}
-Let <math>G</math> be the dihedral group of order eight. Then, every automorphism of <math>G</math> fixes every element of the center of <math>G</math>, and also, the inner automorphism group of <math>G</math> is maximal in the automorphism group of <math>G</math>. Thus, every automorphism of <math>G</math> is normal-extensible.
+Let <math>G</math> be the dihedral group of order eight. Then, every automorphism of <math>G</math> fixes every element of the center of <math>G</math>, and also, the inner automorphism group of <math>G</math> is maximal in the automorphism group of <math>G</math>. Thus, by fact (1), every automorphism of <math>G</math> is normal-extensible.
 However, there is an automorphism of <math>G</math> that interchanges the two normal Klein four-subgroups. Thus, these two normal subgroups are not invariant under this automorphism, and hence, we have an automorphism of <math>G</math> that is normal-extensible but not normal.
+Equivalently, the Klein four-subgroups are examples of normal subgroups that are not normal-extensible automorphism-invariant.
+===Example involving a simple complete group===
+Let <math>S</math> be a simple complete group. In other words, <math>S</math> is a centerless simple group such that every automorphism of <math>S</math> is inner. Let <math>G = S \times S</math>. By fact (2), the automorphism group of <math>G</math> is the wreath product of <math>S</math> with the symmetric group of degree two, which has <math>G</math>, the inner automorphism group, as a subgroup of index two. Moreover, <math>G</math> is centerless. Thus, by fact (1), we get that every automorphism of <math>G</math> is normal-extensible.
+However, the ''coordinate exchange'' automorphism of <math>G</math>, that interchanges the two copies of <math>S</math>, is not a normal automorphism because it interchanges these two normal subgroups. Thus, we have an example of a normal-extensible automorphism that is not normal.
+Equivalently, either of the direct factors is an example of a normal subgroup that is not normal-extensible automorphism-invariant.