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 (ℤ/𝑞ℤ)× and prove a similar result. For all suitably large primes 𝑞 we define 𝑃𝜂 to be the set of primes less than 𝜂𝑞 , viewed naturally as a subset of (ℤ/𝑞ℤ)× . Considering the 𝑘 ‐fold product set 𝑃(𝑘)𝜂={𝑝1𝑝2⋯𝑝𝑘:𝑝𝑖∈𝑃𝜂} , we show that, for 𝜂≫𝑞−1/4+𝜖, there exists a constant 𝑘 depending only on 𝜖 such that 𝑃(𝑘)𝜂=(ℤ/𝑞ℤ)× . Erdös conjectured that, for 𝜂=1, the value 𝑘=2 should suffice: although we have not been able to prove this conjecture, we do establish that 𝑃(2)1 has density at least 164(1+𝑜(1)) . We also formulate a similar theorem in almost‐primes, improving on existing results.