# Characteristic not implies elementarily characteristic

From Groupprops

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) neednotsatisfy the second subgroup property (i.e., elementarily characteristic subgroup)

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## Statement

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.

## Proof

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.