  The classical theorem of Schnirelmann states that the primes are an additive
basis for the integers. In this paper we consider the analogous multiplicative
setting of the cyclic group $\left(\mathbb{Z}/ q\mathbb{Z}\right)^{\times}$,
and prove a similar result. For all suitably large primes $q$ we define
$P_\eta$ to be the set of primes less than $\eta q$, viewed naturally as a
subset of $\left(\mathbb{Z}/ q\mathbb{Z}\right)^{\times}$. Considering the
$k$-fold product set $P_\eta^{(k)}=\{p_1p_2\cdots p_k:p_i\in P_\eta \}$, we
show that for $\eta \gg q^{-\frac{1}{4}+\epsilon}$ there exists a constant $k$
depending only on $\epsilon$ such that $P_\eta^{(k)}=\left(\mathbb{Z}/
q\mathbb{Z}\right)^{\times}$. Erd\H{o}s conjectured that for $\eta = 1$ the
value $k=2$ should suffice: although we have not been able to prove this
conjecture, we do establish that $P_1 ^{(2)}$ has density at least
$\frac{1}{64}(1+o(1))$. We also formulate a similar theorem in almost-primes,
improving on existing results.
