Let G be a locally compact group, and let 1 < p < \infty. In this paper we
investigate the injectivity of the left L^1(G)-module L^p(G). We define a
family of amenability type conditions called (p,q)-amenability, for any 1 <= p
<= q. For a general locally compact group G we show if L^p(G) is injective,
then G must be (p,p)-amenable. For a discrete group G we prove that l^p(G) is
injective if and only if G is (p,p)-amenable.