Direct factor 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., direct factor) need not satisfy the second subgroup property (i.e., characteristic subgroup)
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Similar facts for subgroup properties
- Normal not implies characteristic
- Normal not implies direct factor
- Characteristic not implies direct factor
Analogues in Lie rings
- Direct factor not implies derivation-invariant
- Direct factor not implies characteristic in Lie rings
Example of a direct product
Let be any nontrivial group. Then consider , viz., the external direct product of with itself. The subgroups and are direct factors of , and are hence both normal in . Note also that they are distinct, since is nontrivial.
However, the exchange automorphism:
exchanges the subgroups and . Thus, neither nor is invariant under all the automorphisms, so neither is characteristic. Thus, and are both direct factors of that are not characteristic.