If $G$ is an infinite compact group, how many orbits can $G$ have under the group action of its continuous automorphisms ?
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$\begingroup$ Do you just want the cardinality, or some kind of structure on the equivalence relation induced by the automorphisms? $\endgroup$– Colin ReidCommented Nov 3, 2011 at 11:36
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$\begingroup$ I was first wondering about if their could be more than countably many orbits, but structure results 1) on the group (depending on the number of orbits) or as you say 2) on the equivalence relation induced would be helpful. $\endgroup$– DrikeCommented Nov 4, 2011 at 6:23
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If $G$ is a simple compact matrix Lie group of positive dimension, then there are continuously many orbits: elements with distinct eigenvalues are not conjugate and the outer automorphism group is finite.
If on the other hand $G=\mathbb{Z}_p$ with $p$ a prime, there are countably many orbits.
On yet another hand, if a compact topological group $G$ is assumed infinite, then I don't think the number of the orbits can be finite: if I remember correctly, $G$ must contain a copy of $S^1$ or the $p$-adic integers, so there will always be countably or continuously many orbits.
Note that if we allow for finite groups, then we can have as many orbits as we wish: if $G=\mathbb{Z}/p^n$ with $p$ a prime, then there are $n+1$ orbits.
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$\begingroup$ One more example: for $G=\mathbb{T}^n$ (the $n$-dimensional torus), $Aut(G)=GL_n(\mathbb{Z})$, so there are uncountably many orbits. $\endgroup$ Commented Nov 3, 2011 at 18:18
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$\begingroup$ Still another example: the Cantor group $C=\{0,1\}^\mathbb{N}$; here there are countably many orbits under $Aut(C)$. $\endgroup$ Commented Nov 3, 2011 at 21:35
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$\begingroup$ Many thanks for your answer. Would you have any references where I could check the details (and learn more about Aut(C) when C is a compact group) ? Also, do you think there are essential differences on the structure of such a compact group C depending on whever the set of orbits is countable or uncountable ? Can the set of orbits be of size aleph_1 ? $\endgroup$– DrikeCommented Nov 4, 2011 at 6:33
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$\begingroup$ Drike -- one reference is Hofmann-Morris, The structure of compact groups, De Gruyter Studies in Mathematics 25, in particular chapter 8, the structure theory of abelian compact groups. Every such group has a totally disconnected normal subgroup with quotient a torus (proposition 8.15). The structure of totally disconnected groups is given in ibid., Proposition 8.8. $\endgroup$– algoriCommented Nov 4, 2011 at 16:53