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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)
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Contents 1 Statement 1.1 Verbal statement 1.2 Statement with symbols 2 Related facts 2.1 Stronger facts 2.2 Weaker facts 3 Facts used 4 Proof 4.1 Example of the dihedral group

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

Related facts

Stronger facts

• Normal not implies normal-potentially relatively characteristic
• Potentially characteristic not implies semi-strongly potentially characteristic
• Potentially characteristic not implies semi-strongly potentially relatively characteristic

Weaker facts

• Normal not implies characteristic-potentially characteristic

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.

• H is not a normal-potentially characteristic subgroup of G: Using the fact that every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible, every automorphism of G can be extended to an automorphism of K for any group K containing G as a normal subgroup. But since there is an automorphism of G not sending H to itself, H cannot be characteristic in K.
• H is normal in G: This is obvious.