Normal not implies normal-potentially characteristic: Difference between revisions

Latest revision as of 07:07, 22 February 2013

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-potentially characteristic 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-potentially characteristic subgroup

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Statement

Verbal statement

It is possible to have a normal subgroup of a group that is not a normal-potentially characteristic subgroup.

Statement with symbols

We can have a group $G$ with a subgroup $H$ such that $H$ is normal in $G$ , but whenever $K$ is a group containing $G$ as a normal subgroup, $H$ is not a characteristic subgroup in $K$ .

Related facts

Stronger facts

Normal not implies normal-potentially relatively characteristic

Weaker facts

Normal not implies characteristic-potentially characteristic

Facts used

Proof

The proof follows directly from facts (1) and (2).

Example of the dihedral group

Further information: dihedral group:D8

Let $G$ be the dihedral group of order eight, and $H$ be one of the Klein four-subgroups.

$H$ is not a normal-potentially characteristic subgroup of $G$ : Using the fact that every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible, every automorphism of $G$ can be extended to an automorphism of $K$ for any group $K$ containing $G$ as a normal subgroup. But since there is an automorphism of $G$ not sending $H$ to itself, $H$ cannot be characteristic in $K$ .
$H$ is normal in $G$ : This is obvious.

@@ Line 1: / Line 1: @@
 {{subgroup property non-implication|
 stronger = normal subgroup|
-weaker = semi-strongly potentially characteristic subgroup}}
+weaker = normal-potentially characteristic subgroup}}
 ==Statement==
@@ Line 7: / Line 7: @@
 ===Verbal statement===
-It is possible to have a [[normal subgroup]] of a [[group]] that is not a [[semi-strongly potentially characteristic subgroup]].
+It is possible to have a [[normal subgroup]] of a [[group]] that is not a [[normal-potentially characteristic subgroup]].
 ===Statement with symbols===
-We can have a group <math>K</math> with a subgroup <math>H</math> such that <math>H</math> is normal in <math>K</math>, but whenever <math>G</math> is a group containing <math>K</math> as a [[normal subgroup]], <math>H</math> is ''not'' a [[characteristic subgroup]] in <math>G</math>.
+We can have a group <math>G</math> with a subgroup <math>H</math> such that <math>H</math> is normal in <math>G</math>, but whenever <math>K</math> is a group containing <math>G</math> as a [[normal subgroup]], <math>H</math> is ''not'' a [[characteristic subgroup]] in <math>K</math>.
 ==Related facts==
@@ Line 17: / Line 17: @@
 ===Stronger facts===
-* [[Weaker than::Normal not implies semi-strongly potentially relatively characteristic]]
+* [[Weaker than::Normal not implies normal-potentially relatively characteristic]]
-* [[Weaker than::Potentially characteristic not implies semi-strongly potentially characteristic]]
-* [[Weaker than::Potentially characteristic not implies semi-strongly potentially relatively characteristic]]
 ===Weaker facts===
-* [[Stronger than::Normal not implies strongly potentially characteristic]]
+* [[Stronger than::Normal not implies characteristic-potentially characteristic]]
 ==Facts used==
 # [[uses::Normal not implies normal-extensible automorphism-invariant]]
-# [[uses::Semi-strongly potentially characteristic implies normal-extensible automorphism-invariant]]
+# [[uses::Normal-potentially characteristic implies normal-extensible automorphism-invariant]]
 ==Proof==
 The proof follows directly from facts (1) and (2).
+===Example of the dihedral group===
+{{further|[[Particular example::dihedral group:D8]]}}
+Let <math>G</math> be the dihedral group of order eight, and <math>H</math> be one of the Klein four-subgroups.
+* <math>H</math> is not a normal-potentially characteristic subgroup of <math>G</math>: Using the fact that [[every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible]], every automorphism of <math>G</math> can be extended to an automorphism of <math>K</math> for any group <math>K</math> containing <math>G</math> as a normal subgroup. But since there is an automorphism of <math>G</math> not sending <math>H</math> to itself, <math>H</math> cannot be characteristic in <math>K</math>.
+* <math>H</math> is normal in <math>G</math>: This is obvious.