Normal-extensible not implies normal: Difference between revisions

Revision as of 19:54, 27 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

Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible

Proof

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, 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.

@@ Line 37: / Line 37: @@
 ==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.
 ==Proof==
-{{further|[[Particular example::general linear group:GL(3,2)]]}}
+{{further|[[Particular example::dihedral group:D8]], [[subgroup structure of dihedral group:D8]]}}
-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>.
+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.
-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>.
+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.
-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.
-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.