Potentially characteristic not implies normal-potentially characteristic

From Groupprops
Jump to: navigation, search
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., potentially characteristic subgroup) need not satisfy the second subgroup property (i.e., normal-potentially characteristic subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about potentially characteristic subgroup|Get more facts about normal-potentially characteristic subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property potentially characteristic subgroup but not normal-potentially characteristic subgroup|View examples of subgroups satisfying property potentially characteristic subgroup and normal-potentially characteristic subgroup

Statement

It is possible to have a potentially characteristic subgroup that is not a normal-potentially characteristic subgroup.

Related facts

Weaker facts

Facts used

  1. Potentially characteristic not implies normal-extensible automorphism-invariant, which in turn follows from Normal not implies normal-extensible automorphism-invariant in finite and finite normal implies potentially characteristic.
  2. Normal-potentially characteristic implies normal-extensible automorphism-invariant

Proof

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

Example of the dihedral group

Further information: dihedral group:D8

Let G be the dihedral group of order eight, and H be one of the Klein four-subgroups.