For torsion-free groups it is proved in Kurosh, "Group theory", See 3d edition, Chapter VIII, Section 30 (of course the result can be found in the 1st edition as well). Oroginally it was proved in Reinhold Baer, "Abelian groups without elements of finite order". Duke Math J. 3 (1): 68–122, 1937. For groups with torsion, it follows from old results as well but I am not sure anybody specifically mentioned it somewhere. Of course you need to look at Fuchs, "Infinite Abelian groups" (both volumes). If it is not there, it is probably not anywhere else.
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The fact is true only if the group is For torsion-free . It groups it is proved in Kurosh, "Group theory", See 3d edition, Chapter VIII, Section 30 (of course the result can be found in the 1st edition as well). Oroginally it was proved in Reinhold Baer, "Abelian groups without elements of finite order". Duke Math J. 3 (1): 68–122, 1937. For groups with torsion, it follows from old results as well but I am not sure anybody specifically mentioned it somewhere. |
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The fact is true only if the group is torsion-free. It is proved in Kurosh, "Group theory", See 3d edition, Chapter VIII, Section 30 (of course the result can be found in the 1st edition as well). |
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