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

From Groupprops

(→Verbal statement=) |
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Line 1: | Line 1: | ||

{{subgroup property non-implication| | {{subgroup property non-implication| | ||

stronger = normal subgroup| | stronger = normal subgroup| | ||

− | weaker = | + | weaker = characteristic-potentially characteristic subgroup}} |

==Statement== | ==Statement== | ||

Line 7: | Line 7: | ||

===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=== | ||

Line 15: | Line 15: | ||

==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) neednotsatisfy 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).