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

Let ( G , ? ) be a finite group of order v . A ( G , k , λ ) difference matrix (briefly, ( G , k , λ ) -DM) is a k × λ v matrix D = ( d i j ) with entries from G , so that for any distinct rows x and y , the multiset { d x i ? d y i - 1 : 1 ? i ? λ v } contains each element of G exactly λ times. In this paper, we are concerned about a ( G , 4 , λ ) -DM whatever structure of a finite abelian group G is. Eventually for the following two cases: (1) λ = 1 and G is non-cyclic, (2) λ 1 is an odd integer, we prove that a ( G , 4 , λ ) -DM exists if and only if G has no non-trivial cyclic Sylow 2-subgroups. Moreover, we point out that a ( G , 4 , λ ) -DM always exists for any even integer λ ? 2 and any finite abelian group G .