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Hall's universal group has ample generic automorphisms
Author(s) -
Song Shichang
Publication year - 2019
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201800053
Subject(s) - mathematics , automorphism , conjugacy class , group (periodic table) , outer automorphism group , countable set , diagonal , pure mathematics , automorphisms of the symmetric and alternating groups , discrete mathematics , combinatorics , automorphism group , geometry , chemistry , organic chemistry
We show that the automorphism group of Hall's universal locally finite group has ample generics, that is, it admits comeager diagonal conjugacy classes in all dimensions. Consequently, it has the small index property, is not the union of a countable chain of non‐open subgroups, and has the automatic continuity property. Also, we discuss some algebraic and topological properties of the automorphism group of Hall's universal group. For example, we show that every generic automorphism of Hall's universal group is conjugate to all of its powers, and hence has roots of all orders.