Difference between revisions of "Normal not implies strongly potentially characteristic"

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(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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==Statement==
 
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==Verbal statement===
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===Verbal statement===
  
 
A [[normal subgroup]] need not be [[strongly potentially characteristic subgroup|strongly potentially characteristic]].
 
A [[normal subgroup]] need not be [[strongly potentially characteristic subgroup|strongly potentially characteristic]].

Revision as of 12:58, 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., 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 strongly potentially characteristic subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal subgroup but not strongly potentially characteristic subgroup|View 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 K and a normal subgroup H of K such that there is no group G containing K in which both H and K are characteristic subgroups.

Facts used

  1. Strongly potentially characteristic implies semi-strongly potentially characteristic
  2. Normal not implies semi-strongly potentially characteristic

Proof

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