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@ACL - I see that I hadn't answered your comment. Sorry! I don't remember the details, but I was referring to a paragraph on p. 38 of Weil's book on topological groups. I added a screenshot of the paragraph to the question. I'm afraid I overstated Weil's comment.
@lambda - Thanks a lot! Looks very interesting! If I understand correctly you're saying that if $C$ is a maximal chain in a boolean lattice, then $C$ isn't fixed by any nontrivial automorphism. Could you give me an argument or a reference?
