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

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(Verbal statement=)
 
Line 1: Line 1:
 
{{subgroup property non-implication|
 
{{subgroup property non-implication|
 
stronger = normal subgroup|
 
stronger = normal subgroup|
weaker = strongly potentially characteristic subgroup}}
+
weaker = characteristic-potentially characteristic subgroup}}
  
 
==Statement==
 
==Statement==
Line 7: Line 7:
 
===Verbal statement===
 
===Verbal statement===
  
A [[normal subgroup]] need not be [[strongly potentially characteristic subgroup|strongly potentially characteristic]].
+
A [[normal subgroup]] need not be [[characteristic-potentially characteristic subgroup|characteristic-potentially characteristic]].
  
 
===Statement with symbols===
 
===Statement with symbols===
Line 15: Line 15:
 
==Facts used==
 
==Facts used==
  
# [[uses::Strongly potentially characteristic implies semi-strongly potentially characteristic]]
+
# [[uses::Characteristic-potentially characteristic implies normal-potentially characteristic]]
# [[uses::Normal not implies semi-strongly potentially characteristic]]
+
# [[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 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. Characteristic-potentially characteristic implies normal-potentially characteristic
  2. Normal not implies normal-potentially characteristic

Proof

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