Normal not implies normal-potentially characteristic: Difference between revisions

Revision as of 22:29, 23 April 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., semi-strongly 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 semi-strongly potentially characteristic subgroup

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

Statement

Verbal statement

It is possible to have a normal subgroup of a group that is not a semi-strongly 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 $G$ .

Facts used

Proof

By fact (1), we can find a group $K$ , a normal subgroup $H$ of $K$ , and a normal-extensible automorphism $σ$ of $K$ such that $σ (H) \neq H$ . Thus, for any group $G$ containing $K$ as a normal subgroup, $σ$ extends to an automorphism $σ^{'}$ of $G$ . But then, $σ^{'} (H) = σ (H) \neq H$ , so $H$ is not characteristic in $G$ .

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 ==Facts used==
-# [[uses::Normal-extensible not implies normal]]
+# [[uses::Normal not imples normal-extensible automorphism-invariant]]
+# [[uses::Semi-strongly potentially characteristic implies normal-extensible automorphism-invariant]]
 ==Proof==
 By fact (1), we can find a group <math>K</math>, a normal subgroup <math>H</math> of <math>K</math>, and a normal-extensible automorphism <math>\sigma</math> of <math>K</math> such that <math>\sigma(H) \ne H</math>. Thus, for any group <math>G</math> containing <math>K</math> as a normal subgroup, <math>\sigma</math> extends to an automorphism <math>\sigma'</math> of <math>G</math>. But then, <math>\sigma'(H) = \sigma(H) \ne H</math>, so <math>H</math> is not characteristic in <math>G</math>.