Geodesic
Elementary equivalence and the commutator subgroup
Glasgow mathematical journal, Tome 23 (1982) no. 2, pp. 115-117

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If G and H are elementarily equivalent groups (that is, no elementary statement of group theory distinguishes between G and H) then the definable subgroups of G are elementarily equivalent to the corresponding ones of H. But G′ of G, consisting of all products of commutators [a, b] = a−1b−1ab of elements a and b of G, may not be definable. Must G′ and H′ be elementarily equivalent? 
Dries, L. P. D. van den; Glass, A. M. W.; Mekler, Alan H.; Poland, John. Elementary equivalence and the commutator subgroup. Glasgow mathematical journal, Tome 23 (1982) no. 2, pp. 115-117. doi: 10.1017/S0017089500004870
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