The restricted sumsets in finite abelian groups

Suppose that $k\ge 2$ and $A$ is a non-empty subset of a finite abelian group $G$ with $\mathrm{\mid }G\mathrm{\mid }>1$. Then the cardinality of the restricted sumset $$k^{\wedge }A:=\{a_1+\cdots +a_k:a_1,\dots ,a_k\in A,a_i\mathrm{\ne }{a}_j\text{ for }i\mathrm{\ne }j\}$$ is at least $$\min \{p(G),k\mathrm{\mid }A\mathrm{\mid }-k^2+1\},$$ where $p(G)$ denotes the least prime divisor of $\mathrm{\mid }G\mathrm{\mid }$.