Normal-extensible not implies normal: Difference between revisions

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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 2: / Line 2: @@
 stronger = normal-extensible automorphism|
 weaker = normal automorphism}}
+{{subgroup property non-implication|
+stronger = normal subgroup|
+weaker = 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]] <math>G</math> and a [[normal-extensible automorphism]] <math>\sigma</math> of <math>G</math> that is ''not'' a [[normal automorphism]]: in other words, there exists a [[normal subgroup]] <math>N</math> of <math>G</math> such that <math>\sigma(N) \ne N</math>.
+==Related facts==
+* [[Centerless and maximal in automorphism group implies every automorphism is normal-extensible]]
+* [[Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible]]
+* [[Normal-extensible not implies inner]]
+* [[Normal-extensible not implies extensible]]
+===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===
+* [[Normal not implies semi-strongly potentially characteristic]]
+* [[Normal not implies strongly potentially characteristic]]
 ==Facts used==
-# [[uses::Centerless and maximal in automorphism group implies every automorphism is normal-extensible]]
+# [[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 is wreath product of automorphism group and symmetric group]]: Let <math>S</math> be a simple non-abelian group and <math>S^n</math> be a direct product of <math>S</math> with itself <math>n</math> times. The automorphism group of <math>S^n</math> is <math>\operatorname{Aut}(S) \wr \operatorname{Sym}(n)</math>, where the latter has the usual action on <math>n</math> letters.
+# [[uses::Automorphism group of direct power of simple non-abelian group equals wreath product of automorphism group and symmetric group]]
 ==Proof==
-{{further|[[Particular example::general linear group:GL(3,2)]]}}
+===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, 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.
-Let <math>S = GL(3,2)</math>, i.e., <math>S</math> is the general linear group of order three over the field of two elements. Consider <math>G = S \times S</math>.
+===Example involving a simple complete group===
-We know that <math>S</math> is simple and non-abelian (hence centerless) and every automorphism of <math>S</math> is inner, so <math>S = \operatorname{Aut}(S)</math>.
+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.
-By fact (2), the automorphism group of <math>G</math> is <math>\operatorname{Aut}(S) \wr \mathbb{Z}/2\mathbb{Z}</math>. Because every automorphism of <math>S</math> is inner, this is <math>G \rtimes \mathbb{Z}/2\mathbb{Z}</math>, with the action being the coordinate exchange automorphism. Thus, <math>G</math> is centerless and maximal in its automorphism, so by fact (1), 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.
-Let <math>N</math> be the normal subgroup of <math>G</math> defined by the first direct factor, i.e., <math>N = S \times 1</math>. Let <math>\sigma</math> be the coordinate exchange automorphism of <math>G</math>, i.e., <math>\sigma(a,b) = (b,a)</math>. Then, since every automorphism of <math>G</math> is normal-extensible, <math>\sigma</math> is normal-extensible. However, <math>\sigma(N) \ne N</math>, since <math>\sigma(N) = 1 \times S</math> is the other direct factor.
+Equivalently, either of the direct factors is an example of a normal subgroup that is not normal-extensible automorphism-invariant.