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 need not satisfy the second subgroup property
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
|

Property "Page" (as page type) with input value "{{{stronger}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.Property "Page" (as page type) with input value "{{{weaker}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property {{{stronger}}} but not {{{weaker}}}|View examples of subgroups satisfying property {{{stronger}}} and {{{weaker}}}

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)