Characteristic not implies isomorph-free in finite group
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a finite group, every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) need not satisfy the second subgroup property (i.e., isomorph-free subgroup)
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Statement with symbols
- Characteristic not implies isomorph-containing
- Characteristic not implies sub-isomorph-free in finite
- Characteristic not implies isomorph-normal in finite
- Characteristic not implies sub-(isomorph-normal characteristic) in finite
- Characteristic not implies injective endomorphism-invariant
Example of the dihedral group
Let be the dihedral group of order eight, given as follows, where denotes the identity element of :
Let be the center of . is a subgroup of order two generated by .
- is characteristic.
- is not isomorph-free: The subgroup of is isomorphic to .