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Let A be a multiplicative subgroup of $\mathbb{Z}{*}_p$. Define the k-fold sumset of A to be $kA=x_1+...+x_k:x_i\in A,1\le i\le k$. We show that $6A\supseteq \mathbb{Z}{*}_p$ for $\left|A\right|>p^{11/23+ϵ}$. In addition, we extend a result of Shkredov to show that ${\left|2A\right|\gg \left|A\right|}^{8/5-ϵ}$ for $\left|A\right|\ll p^{5/9}$.

Let p be a prime, ℤₚ be the finite field in p elements, k be a positive integer, and A be the multiplicative subgroup of nonzero kth powers in ℤₚ. The goal of this paper is to determine, for a given positive integer s, a value tₛ such that if |A| ≫ tₛ then every element of ℤₚ is a sum of s kth powers. We obtain $t₄=p^{22/39+ϵ}$, $t₅=p^{15/29+ϵ}$ and for s ≥ 6, $tₛ=p^{(9s+45)/(29s+33)+ϵ}$. For s ≥ 24 further improvements are made, such as $t_{32}=p^{5/16+ϵ}$ and $t_{128}=p^{1/4}$.