  We generalize two results about subgroups of multiplicative group of finite
field of prime order. In particular, the lower bound on the cardinality of the
set of values of polynomial $P(x,y)$ is obtained under the certain conditions,
if variables $x$ and $y$ belong to a subgroup $G$ of the multiplicative group
of the filed of residues. Also the paper contains a proof of the result that
states that if a subgroup $G$ can be presented as a set of values of the
polynomial $P(x,y)$, where $x\in A$, and $y\in B$ then the cardinalities of
sets $A$ and $B$ are close (in order) to a square root of the cardinality of
subgroup $G$.
