Characteristic not implies elementarily 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., characteristic subgroup) need not satisfy the second subgroup property (i.e., elementarily characteristic subgroup)
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It is possible to have a characteristic subgroup of a group that is not an elementarily characteristic subgroup of , i.e., there exists a subgroup of such that the first-order theories of the group-subgroup pairs and are elementarily equivalent.
Suppose are two infinite sets with |T_1| > |T_2|</math>. Let be the direct product where is the finitary alternating group on and is the finitary alternating group on . Let be the subgroup of that is the first direct factor () and be the subgroup of that is the second direct factor ().
- is characteristic (in fact, homomorph-containing) in : By cardinality considerations and the fact that is simple, any homomorphic image of in is trivial. Therefore, the image of under any homomorphism of is contained in . Therefore, is characteristic in .
- The first-order theories of the group-subgroup pairs and are elementarily equivalent, so is not elementarily characteristic in : This can be seen from the idea that first-order statements reference only finite subsets, and as far as finite subsets are concerned, and look the same.