Let $G$ be a infinite abelian group . We know that we can determine structure the automorphisms group of all finite abelian groups.
Can we determine structuer the automorphisms group of all infinite abelian groups?
Thank you
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$\begingroup$ Short answer: No. $\endgroup$– Martin BrandenburgApr 11, 2013 at 18:29
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$\begingroup$ Perhaps we should first ask for a classification of all abelian groups, before we want to completely determine all their automorphism groups. $\endgroup$– anonApr 11, 2013 at 18:45
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$\begingroup$ Infinite abelian groups can be beasts covered by a lamb skin...beware of them! $\endgroup$– DonAntonioApr 11, 2013 at 18:48
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$\begingroup$ You say "We know that we can determine (the) structure (of) the automorphism group of all finite abelian groups". However, according to this paper, the automorphism groups of finite abelian groups seem to have passed the literature by. (The paper does give a complete characterisation of them though.) $\endgroup$– user1729Apr 12, 2013 at 10:53
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1 Answer
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We don't even know what all the infinite abelian groups are, much less do we have a description of their automorphism groups. This question has some information about that.
Finitely generated abelian groups we have a better knowledge of, in particular of tame automorphisms, but these are not completely classified either. In fact, even the automorphism groups of finite abelian groups are hard to describe in general.
answered Apr 11, 2013 at 19:12
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$\begingroup$ By tame automorphisms, do you mean those automorphisms which lift to automorphisms of the ambient free group? $\endgroup$– user1729Apr 12, 2013 at 9:25