Let G be a connected, simply connected nilpotent Lie group and S (G) the space of the discrete uniform subgroups of G. In this paper, we give some necessary and sufficient conditions for S (G) under which the group G is abelian. In particular, we prove the non-existence of nonabelian nilpotent Lie groups with one discrete uniform subgroup up to automorphism.Mathematics Subject Classification. Primary 22E40.
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