Difference between revisions of "Normal not implies strongly potentially characteristic"
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(Created page with '{{subgroup property nonimplication stronger = normal subgroup weaker = strongly potentially characteristic subgroup}} ==Statement== ==Verbal statement=== A [[normal subgrou...') 
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Revision as of 12:57, 23 April 2009
This article gives the statement and possibly, proof, of a nonimplication 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., strongly potentially characteristic subgroup)
View a complete list of subgroup property nonimplications  View a complete list of subgroup property implications
Get more facts about normal subgroupGet more facts about strongly potentially characteristic subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal subgroup but not strongly potentially characteristic subgroupView examples of subgroups satisfying property normal subgroup and strongly potentially characteristic subgroup
Statement
Verbal statement=
A normal subgroup need not be strongly 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
 Strongly potentially characteristic implies semistrongly potentially characteristic
 Normal not implies semistrongly potentially characteristic
Proof
The proof follows directly from facts (1) and (2).