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
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Statement
A normal subgroup of a group need not be a characteristic subgroup.
Example
Let be any nontrivial group. Then consider , viz the external direct product of with itself. The subgroup is a normal subgroup of (being one of the direct factors).
However, is not a characteristic subgroup, because it is not invariant under the automorphism (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. HersteinMore info, Page 70 (Problem 7(a))
- Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347More info, Page 137 (Problem 6)