Any small multiplicative sugroup is not a sumset

We prove that for an arbitrary $\epsilon >0$ and any multiplicative subgroup $\mathrm{\Gamma }\subseteq {\mathbf{F}}_p$, $1\ll |\Gamma| \le p^{2/3 -\varepsilon}$therearenosets$B$,$C \subseteq $\mathbf{F}$_p$with$|B|, |C|>1$suchthat$\Gamma=B+C$\mathrm{.}Also,weobtainthatfor$1\ll |\Gamma| \le p^{6/7-\varepsilon}$andany$\xi\neq 0$thereisnoaset$B$suchthat$\xi \Gamma+1=B/B$.