In this paper we consider the unimodular solvable Lie group 1 G n. As it is stated in [9], in 1980, Bozek has introduced G n for the first 2 time. In [9] Calvaruso, Kowalski and Marinosci have studied geodesics 3 on this Lie group when it has arbitrary odd dimension. Our aim in this 4 paper is to investigate four other geometrical properties i.e. homogeneous 5 Ricci solitons, harmonicity of invariant vector fields, left invariant con-6 tact structures and homogeneous structures in two cases

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... annian and 7 Lorentzian on this Lie group with dimension 5. This survey shows that, 8 the space-like energy on the Lorentzian Lie group G 2 does not have a 9 critical point and there is no left invariant almost complex structure on 10 G 2 × R. 11 M.S.C. 2010: 53C50, 53C43, 53C15, 53C30. 12