Characteristic not implies elementarily characteristic

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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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Statement

It is possible to have a characteristic subgroup of a group G that is not an elementarily characteristic subgroup of G.

Proof

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