Suppose $p$ is prime. What can be said about the structure and the size of a set $ A \subseteq F^*_p$ ($F^*_p $is a multiplicative group of integers modulo $p $), provided that it does not change under multiplication by a multiplicative subgroup $ G \subseteq F^*_p$, i.e.
$$ GA = \{x*a: x \in G, a \in A\} = A $$
I tried representing $ G $ as a set of successive powers of a primitive root module $ p $, and I guess that the set satisfying this condition is a larger multiplicative subgroup of $ F^*_p $. Is this correct?
In general, for any group $U$, let $G$ be a subgroup of $U$ and let $A$ be any subset of $U$. It is easy to check that the condition $GA=A$ is equivalent to: $A$ is a union of right cosets of $G$. This includes "$A$ is a subgroup of $U$ containing $G$" but is broader.
For example, let $p=7$, and let $G=\{1,6\}$ (considering $F_p \cong \Bbb Z/7\Bbb Z$ to be represented by $\{0,1,2,3,4,5,6\}$). Note that cosets of $G$ in $F_p^*$ are $\{1,6\}$, $\{2,5\}$, and $\{3,4\}$. Then besides $A=G$ and $A=F_p^*$, there are two other examples for which $GA=A$, namely $$ A = \{1,2,5,6\} \quad\text{and}\quad A = \{1,3,4,6\}. $$