Normal not implies characteristic

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 subgroup)
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Statement

A Normal subgroup (?) of a group need not be a Characteristic subgroup (?).

Example

Let $G$ be any nontrivial group. Then consider $K=G\times G$, viz the external direct product of $G$ with itself. The subgroup $G\times \{e\}$ is a normal subgroup of $K$ (being one of the direct factors).

However, $G\times \{e\}$ is not a characteristic subgroup, because it is not invariant under the automorphism $(x,y)\mapsto (y,x)$ (called the exchange automorphism).

Note that this example also shows that direct factor does not imply characteristic subgroup.

Partial truth

Groups where every normal subgroup is characteristic

A group in which every normal subgroup is characteristic is termed a N=C-group.

Converse

Further information: Characteristic implies normal

The converse statement is indeed true. That is, every characteristic subgroup is normal.

References

Textbook references

• Topics in Algebra by I. N. Herstein^{More info}, Page 70 (Problem 7(a))
• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347^{More info}, Page 137 (Problem 6)