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An Algebraic Obstruction to Isomorphism of Markov Shifts with Group Alphabets
Author(s) -
Ward T. B.
Publication year - 1993
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/25.3.240
Subject(s) - mathematics , abelian group , isomorphism (crystallography) , combinatorics , cardinality (data modeling) , group (periodic table) , discrete mathematics , algebraic number , g module , free abelian group , elementary abelian group , pure mathematics , rank of an abelian group , mathematical analysis , chemistry , organic chemistry , computer science , crystal structure , data mining , crystallography
Given a compact group G , a standard construction of a Z 2 Markov shift ∑ G with alphabet G is described. The cardinality of G (if G is finite) or the topological dimension of G (if G is a torus) is shown to be an invariant of measurable isomorphism for ∑ G . We show that if G is sufficiently non‐abelian (for instance A 5 , PSL 2 (F 7 ) or a Suzuki simple group) and H is any abelian group with ∣ H ∣ = ∣ G ∣, then ∑ G and ∑ H are not isomorphic. Thus the cardinality of G is seen to be necessary but not sufficient to determine the measurable structure of ∑ G .