Normal not implies normal-potentially characteristic

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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., normal-potentially characteristic subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
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Statement

Verbal statement

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

Statement with symbols

We can have a group G with a subgroup H such that H is normal in G, but whenever K is a group containing G as a normal subgroup, H is not a characteristic subgroup in K.

Related facts

Stronger facts

Weaker facts

Facts used

  1. Normal not implies normal-extensible automorphism-invariant
  2. Normal-potentially characteristic implies normal-extensible automorphism-invariant

Proof

The proof follows directly 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.