Normal not implies normal-potentially characteristic: Difference between revisions

Revision as of 21:31, 30 May 2009

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

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal subgroup but not normal-potentially characteristic subgroup|View examples of subgroups satisfying property normal subgroup and normal-potentially characteristic subgroup

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

Related facts

Stronger facts

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 33: / Line 33: @@
 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.