Some finiteness properties of groups and their automorphism groups

For any cardinal number oc such that 1 less than or equal to alpha less than or equal to c = \R\, we prove the existence of an abelian group H which is simultaneously hopfian and cohopfian and satisfies \H\ = alpha. We introduce the concept of Gamma -separability of a group G for any subgroup Gamma of Inn (G) and generalize Stebe's results on conjugacy separability. We also introduce the analog of property A of Grossman relative to Gamma. If Gamma subset of or equal to Inn(G) is a normal subgroup of Aut(G), where G is finitely generated and Gamma -separable, and possesses property A relative to Gamma, we show that Aut (G)/Gamma is residually finite. For a free group F, we show that property A is valid with respect to any subgroup Gamma of Inn (F). We also verify Gamma -separability of F for a certain class of normal subgroups Gamma of Inn (F).