  We obtain a series of estimates on the number of small integers and small
order Farey fractions which belong to a given coset of a subgroup of order $t$
of the group of units of the residue ring modulo a prime $p$, in the case when
$t$ is small compared to $p$. We give two applications of these results: to the
simultaneous distribution of two high degree monomials $x^{k_1}$ and $x^{k_2}$
modulo $p$ and to a question of J.Holden and P.Moree on fixed points of the
discrete logarithm.
