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Journal of the London Mathematical Society 1986 s2-33(1):49-58; doi:10.1112/jlms/s2-33.1.49
© 1986 by London Mathematical Society
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© Oxford University Press

Groups Embeddable in the Autohomeomorphisms of Q

Alan H. Mekler

Department of Mathematics, Simon Fraser University Burnaby, British Columbia V5A 1S6, Canada

LetH(Q) denote the group of autohomeomorphisms of Q. Suppose that (G, X) is a permutation group on a countably infinite set. Then (G, X) is embeddable in H (Q), if some one-to-one and onto map from Xto Q induces an embedding of G into H(Q). A straightforward characterization of the countable groups embeddable in H(Q) is given. No such characterization exists for all groups. There are independence results for uncountable groups of cardinality less than the continuum. Also a negative answer is given to a question of P. Neumann concerning which countable groups can be embedded in H(Q).


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