A note on difference matrices over non-cyclic finite abelian groups

Let (G,⋅)(G,⋅) be a finite group of order vv. A (G,k,λ)(G,k,λ) difference matrix (briefly, (G,k,λ)(G,k,λ)-DM) is a k×λvk×λv matrix D=(dij)D=(dij) with entries from GG, so that for any distinct rows xx and yy, the multiset {dxi⋅dyi−1:1≤i≤λv} contains each element of GG exactly λλ times. In this paper, we are concerned about a (G,4,λ)(G,4,λ)-DM whatever structure of a finite abelian group GG is. Eventually for the following two cases: (1) λ=1λ=1 and GG is non-cyclic, (2) λ>1λ>1 is an odd integer, we prove that a (G,4,λ)(G,4,λ)-DM exists if and only if GG has no non-trivial cyclic Sylow 2-subgroups. Moreover, we point out that a (G,4,λ)(G,4,λ)-DM always exists for any even integer λ≥2λ≥2 and any finite abelian group GG.