Normal not implies normal-potentially characteristic: Difference between revisions
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{{subgroup property non-implication| | {{subgroup property non-implication| | ||
stronger = normal subgroup| | stronger = normal subgroup| | ||
weaker = | weaker = normal-potentially characteristic subgroup}} | ||
==Statement== | ==Statement== | ||
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===Verbal statement=== | ===Verbal statement=== | ||
It is possible to have a [[normal subgroup]] of a [[group]] that is not a [[ | It is possible to have a [[normal subgroup]] of a [[group]] that is not a [[normal-potentially characteristic subgroup]]. | ||
===Statement with symbols=== | ===Statement with symbols=== | ||
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===Stronger facts=== | ===Stronger facts=== | ||
* [[Weaker than::Normal not implies | * [[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 characteristic]] | ||
* [[Weaker than::Potentially characteristic not implies semi-strongly potentially relatively characteristic]] | * [[Weaker than::Potentially characteristic not implies semi-strongly potentially relatively characteristic]] | ||
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===Weaker facts=== | ===Weaker facts=== | ||
* [[Stronger than::Normal not implies | * [[Stronger than::Normal not implies characteristic-potentially characteristic]] | ||
==Facts used== | ==Facts used== | ||
# [[uses::Normal not implies normal-extensible automorphism-invariant]] | # [[uses::Normal not implies normal-extensible automorphism-invariant]] | ||
# [[uses:: | # [[uses::Normal-potentially characteristic implies normal-extensible automorphism-invariant]] | ||
==Proof== | ==Proof== | ||
The proof follows directly from facts (1) and (2). | The proof follows directly from facts (1) and (2). | ||
Revision as of 18:25, 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 with a subgroup such that is normal in , but whenever is a group containing as a normal subgroup, is not a characteristic subgroup in .
Related facts
Stronger facts
- Normal not implies normal-potentially relatively characteristic
- Potentially characteristic not implies semi-strongly potentially characteristic
- Potentially characteristic not implies semi-strongly potentially relatively characteristic
Weaker facts
Facts used
- Normal not implies normal-extensible automorphism-invariant
- Normal-potentially characteristic implies normal-extensible automorphism-invariant
Proof
The proof follows directly from facts (1) and (2).