Normal not implies strongly potentially characteristic: Difference between revisions
(Created page with '{{subgroup property non-implication| stronger = normal subgroup| weaker = strongly potentially characteristic subgroup}} ==Statement== ==Verbal statement=== A [[normal subgrou...') |
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{{subgroup property non-implication| | {{subgroup property non-implication| | ||
stronger = normal subgroup| | stronger = normal subgroup| | ||
weaker = | weaker = characteristic-potentially characteristic subgroup}} | ||
==Statement== | ==Statement== | ||
==Verbal statement=== | ===Verbal statement=== | ||
A [[normal subgroup]] need not be [[ | A [[normal subgroup]] need not be [[characteristic-potentially characteristic subgroup|characteristic-potentially characteristic]]. | ||
===Statement with symbols=== | ===Statement with symbols=== | ||
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==Facts used== | ==Facts used== | ||
# [[uses:: | # [[uses::Characteristic-potentially characteristic implies normal-potentially characteristic]] | ||
# [[uses::Normal not implies | # [[uses::Normal not implies normal-potentially characteristic]] | ||
==Proof== | ==Proof== | ||
The proof follows directly from facts (1) and (2). | The proof follows directly from facts (1) and (2). | ||
Latest revision as of 18:47, 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., characteristic-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 characteristic-potentially characteristic subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal subgroup but not characteristic-potentially characteristic subgroup|View examples of subgroups satisfying property normal subgroup and characteristic-potentially characteristic subgroup
Statement
Verbal statement
A normal subgroup need not be characteristic-potentially characteristic.
Statement with symbols
It is possible to have a group and a normal subgroup of such that there is no group containing in which both and are characteristic subgroups.
Facts used
- Characteristic-potentially characteristic implies normal-potentially characteristic
- Normal not implies normal-potentially characteristic
Proof
The proof follows directly from facts (1) and (2).