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54 A. Bartels, S. Echterhoff, and W. Lückto g j i;j .g i /j1 .H /. Then the following diagram commutes:colim i2I H 1i .H /n.fg/colim i2I ind G i1.H /iŠcolim i2I H G in .G i =i 1 .H //t H n .fg/Šcolim i2I H G in .i colim i2I H G in .k i /.G=H //Hn H .fg/ind G HŠt G n .G=H / HGn .G=H /,where the horizontal maps are the isomorphism given by induction. For the directedsystem fi 1 .H / j i 2 I g with structure maps i;j j 1i .H / W i 1 .H / !j 1 .H /,the group homomorphism colim i2I i j 1i .H / W colim i2I i1 .H / ! H is an isomorphism.This follows by inspecting the standard model for the colimit over a directedsystem of groups. Hence the left vertical arrow is bijective since H ‹ is strongly continuousby assumption. Therefore it remains to show that the mapcolim i2I H G in .k i/W colim i2I H G in .G i= 1 .H // !colim i2I H G iin . iis surjective.Notice that the map given by the direct sum of the structure mapsMH G in . i G=H / ! colim i2I H G in . i G=H /i2IG=H / (3.5)is surjective. Hence it remains to show for a fixed i 2 I that the image of the structuremapH G in . i G=H / ! colim i2I H G in . i G=H /is contained in the image of the map (3.5).We have the decomposition of the G i -seti G=H into its G i-orbitsaG i = i 1 .gHg 1 / Š ! i G=H; g i 1 .gHg 1 / 7! i .g i /gH:G i .gH /2G i n. i G=H /It induces an identification of ƒ-modulesMH G in G i = 1 .gHg 1 / D H G iG i .gH /2G i n. i G=H /iin . iG=H /:Hence it remains to show for fixed elements i 2 I and G i .gH / 2 G i n.i G=H / thatthe obvious compositionH G in G i = 1 .gHg 1 / H G in . i G=H / ! colim i2I H G in . i G=H /i