A note on sumsets of subgroups in ${\mathbb{Z}}_p^{\ast }$

Let $A$ be a multiplicative subgroup of ${\mathbb{Z}}_p^{\ast }$. Define the $k$-fold sumset of $A$ to be $kA=\{x_1+\cdots +x_k:x_i\in A,1\le i\le k\}$. We show that $6A\supseteq {\mathbb{Z}}_p^{\ast }$ for $\mathrm{\mid }A\mathrm{\mid }>p^{\frac{11}{23}+ϵ}$. In addition, we extend a result of Shkredov to show that $|2A|\gg |A|^{\frac 85-\epsilon}$for$|A|\ll p^{\frac 59}$.