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For any finite abelian group 
$G$ with 
$|G|=m$, 
$A\subseteq G$ and 
$g\in G$, let 
$R_{A}(g)$ be the number of solutions of the equation 
$g=a+b$, 
$a,b\in A$. Recently, Sándor and Yang [‘A lower bound of Ruzsa’s number related to the Erdős–Turán conjecture’, Preprint, 2016, arXiv:1612.08722v1] proved that, if 
$m\geq 36$ and 
$R_{A}(n)\geq 1$ for all 
$n\in \mathbb{Z}_{m}$, then there exists 
$n\in \mathbb{Z}_{m}$ such that 
$R_{A}(n)\geq 6$. In this paper, for any finite abelian group 
$G$ with 
$|G|=m$ and 
$A\subseteq G$, we prove that (a) if the number of 
$g\in G$ with 
$R_{A}(g)=0$ does not exceed 
$\frac{7}{32}m-\frac{1}{2}\sqrt{10m}-1$, then there exists 
$g\in G$ such that 
$R_{A}(g)\geq 6$; (b) if 
$1\leq R_{A}(g)\leq 6$ for all 
$g\in G$, then the number of 
$g\in G$ with 
$R_{A}(g)=6$ is more than 
$\frac{7}{32}m-\frac{1}{2}\sqrt{10m}-1$.
