In this thesis, we present an expanded account of the work done by Dales, Daws, Pham and Ramsden. Let G be a locally compact group, and let L 1 (G) be the group algebra of G. H. G. Dales and M. E. Polyakov have recently investigated when various canonical modules over L 1 (G) have certain well- known homological properties. For example, it was proved that L 1 (G) is injective in L 1 (G)-mod if and only if G is discrete and amenable, and that L 1 (G) is injective in L 1 (G)-mod for every locally compact group G. One of the more difficult questions that they considered seems to have been to characterize the locally compact groups G such that the Banach left L 1 (G)-module L p (G) is injective (for 1 p ?). ince L p (G) is a dual Banach L 1 (G)-module, it follows from that L p (G) is an Injective Banach left L 1 (G)-module whenever G is amenable as a locally compact group; the Converse has been an open problem for a long time. They obtained a partial converse to this theorem in the case where G is discrete. Indeed, they showed that if L p (G) is an injective Banach left L 1 (G)-module for some p€(1,?), then G must be ”pseudo-amenable”, a property very close to amenability. (In fact, no example of a group that is pseudo-amenable, but not amenable, is known.)