For a second countable locally compact abelian group G, we study a system of translates generated by f∈L2(G). We find some equivalent conditions of this family to have some fundamental frame properties. More precisely, let Γ be a uniform lattice in G (a closed subgroup which is cocompact and discrete) and Γ∗ be the annihilator of Γ in Gˆ. For f∈L2(G), the Γ∗-periodic function Φf is defined as Φf(ξ)=∑γ∈Γ∗|fˆ(ξ+γ)|2 on Γˆ (the dual group of Γ) and some of its properties are investigated. In particular, it is shown that if Φf is continuous, then the family {f(.+γ)}γ∈Γ cannot be a redundant frame. Among other things, it is shown that there is an isometry from L2(G) into L2(Γˆ) in such a way that the system of translates in L2(Γˆ) is transferred to a nice Fourier-like system in L2(Γˆ). Also, the canonical and oblique duals of the frames of translates are investigated.

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