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56 A. Bartels, S. Echterhoff, and W. LückProof. The proof is completely analogous to the one in [4, Theorem 2.4, Lemma 2.2],where only the case Df1g is treated.Theorem 4.4. Let .G; / be a group over . Let F be a family of subgroups of G.(i) Let G be the directed union of subgroups fG i j i 2 I g. Suppose that H ‹ iscontinuous and for every i 2 I the isomorphism conjecture holds for .G i ;j Gi /and F j Gi .Then the isomorphism conjecture holds for .G; / and F .(ii) Let fG i j i 2 I g be a directed system of groups with G D colim i2I G i andstructure maps i W G i ! G. Suppose that H ‹ is strongly continuous and forevery i 2 I the isomorphism conjecture holds for .G i ; ı i / andi F .Then the isomorphism conjecture holds for .G; / and F .Proof. (i) The proof is analogous to the one in [4, Proposition 3.4].(ii) This follows from the following commutative square whose horizontal arrowsare bijective because of Lemma 3.4 and the identification i E F .G/ D E i F .G i /:colim i2I H G in.E i F .G i // t G n .E F .G//ŠHn G.E F .G//colim i2I H G in .fg/tn G.fg/Š HGn .fg/.Fix a class of groups C closed under isomorphisms, taking subgroups and takingquotients, e.g., the class of finite groups or the class of virtually cyclic groups. For agroup G let C.G/ be the family of subgroups of G which belong to C.Theorem 4.5. Let .G; / be a group over .(i) Let G be the directed union G D S i2I G i of subgroups G i Suppose that H ‹ iscontinuous and that the isomorphism conjecture is true for .G i ;j Gi / and C.G i /for all i 2 I .Then the isomorphism conjecture is true for .G; / and C.G/.(ii) Let fG i j i 2 I g be a directed system of groups with G D colim i2I G i andstructure maps i W G i ! G. Suppose that H ‹ is strongly continuous and thatthe isomorphism conjecture is true for .H; C.H // for every i 2 I and everysubgroup H G i .Then for every subgroup K G the isomorphism conjecture is true for .K; j K /and C.K/.