We give a refinement of the stable Snaith splitting of the double loop space of a Moore space and use it to construct infinite $v_1$-periodic families of elements of order $p^{ r+1}$ in the homotopy groups of mod $p^r$ Moore spaces. For odd primes $p$, our splitting implies that the homotopy groups of the mod $p^{ r+1}$ Moore spectrum are summands of the unstable homotopy groups of each mod $p^r$ Moore space.
Categories: math.AT