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Groups Embeddable in the Autohomeomorphisms of Q
Author(s) -
Mekler Alan H.
Publication year - 1986
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-33.1.49
Subject(s) - countable set , uncountable set , mathematics , cardinality (data modeling) , combinatorics , characterization (materials science) , group (periodic table) , permutation group , discrete mathematics , permutation (music) , embedding , computer science , physics , acoustics , quantum mechanics , artificial intelligence , optics , data mining
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 X to 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).