Characteristic not implies injective endomorphism-invariant
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., characteristic subgroup) need not satisfy the second subgroup property (i.e., injective endomorphism-invariant subgroup)
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Statement with symbols
It is possible to have a group with a characteristic subgroup that is not injective endomorphism-invariant: in other words, every automorphism of sends to itself, but every injective endomorphism of does not send to itself.
- Finitary symmetric group is characteristic in symmetric group
- Finitary symmetric group is not injective endomorphism-invariant in symmetric group
- Center is characteristic
- Center not is injective endomorphism-invariant
Example of the finitary symmetric group
Let be an infinite set. Let be the symmetric group on , and let be the subgroup comprising finitary permutations. Then, is characteristic in (fact (1)) but is not I-characteristic in (fact (2)).
Example of the center
Facts (3) and (4) give another kind of example: the center of a group is always characteristic, but need not be injective endomorphism-invariant.