A homological characterization of locally finite groups

We show that a group G is locally finite if and only if for all ZG-modules M and I, with M Z-free and I injective. To that end, we relate the finiteness of a finitely generated group G to the finiteness of the injective dimension of induced modules. We also examine the relation between the finiteness of a group G and the finiteness of the projective dimension of coinduced modules.