Normal not implies amalgam-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., amalgam-characteristic subgroup)
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Statement with symbols
- Characteristic not implies amalgam-characteristic: This is a stronger fact, and examples of this also serve as examples of normal not implying amalgam-characteristic.
- Direct factor not implies amalgam-characteristic
- Cocentral not implies amalgam-characteristic
- Finite normal implies amalgam-characteristic
- Central implies amalgam-characteristic
- Normal subgroup contained in hypercenter is amalgam-characteristic
Example of the free group
Let be a free group on two generators and be the group of integers. Let and be the embedded first direct factor. We have:
Thus, is a direct product of two copies of the free group on two generators, and moreover, the embedded subgroup in is simply , the first embedded direct factor. This is not a characteristic subgroup in , because there exists an exchange automorphism swapping the two direct factors of .