Normal not implies normal-potentially characteristic: Difference between revisions

Revision as of 22:30, 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 $G$ .

Related facts

Stronger facts

Weaker facts

Normal not implies strongly potentially characteristic

Facts used

Proof

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

@@ Line 12: / Line 12: @@
 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>.
+==Related facts==
+===Stronger facts===
+* [[Weaker than::Normal not implies semi-strongly 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]]
 ==Facts used==
-# [[uses::Normal not imples normal-extensible automorphism-invariant]]
+# [[uses::Normal not implies 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>.
+The proof follows directly from facts (1) and (2).