Two groups, and more generally two structures, M,N are elementarily equivalent if they satisfy the same first-order sentences, or equivalently, if they have isomorphic ultrapowers. We say that M is an elementary submodel of N, or that N is an elementary extension of M, if M is a substructure of N and if M and N satisfy the same first-order sentences with parameters in M. For each integer n, the sentences with n alternations of quantifiers are the sentences (Q(1)v(1))...(Q(r)v(r))φ with φ a quantifier-free formula, Q(1),...,Q(r) quantifiers and  v(1),...,v(r) variables, for which there exist n integers i such that Q(i)≠Q(i+1).
    A group G is torsion-free if, for each xЄG and each integer n≥1, x^n=1 implies x=1.
    For each group G, consider the subgroups Z(k,G) with Z(0,G)={1} and Z(k+1,G)/Z(k,G) center of G/Z(k,G) for k≥1. Then G is nilpotent of class c≥1 if we have  Z(c,G)=G and Z(c-1,G)≠G.
    A group G is polycyclic if there exists a sequence of subgroups {1}=G(0)≤G(1)≤ ...≤G(n)=G with G(i) normal in G(i+1) and G(i+1)/G(i) cyclic for 0≤i≤n-1. Any finitely generated nilpotent group is polycyclic.
    For any properties P,Q, a group G is P-by-Q if there exists a normal subgroup H of G which satisfies P and such that G/H satisfies Q. Any finitely generated finite-by-abelian group is abelian-by-finite, and any finitely generated finite-by-nilpotent group is nilpotent-by-finite.