Closed Hulls in Infinite Symmetric Groups

Let Sym M be the symmetric group of an infinite set M. What is the smallest subgroup of Sym M containing a given element if the subgroup is subject to the further condition that it is also the automorphism group of some finitary algebra on M? The structures of such closed hulls are related to the di...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 1973-01, Vol.180, p.475-484
Main Author: Haimo, Franklin
Format: Article
Language:English
Subjects:
Algebra
Automorphisms
Direct products
Infinite sets
Integers
Mathematical congruence
Mathematical theorems
Universal algebra
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Summary:Let Sym M be the symmetric group of an infinite set M. What is the smallest subgroup of Sym M containing a given element if the subgroup is subject to the further condition that it is also the automorphism group of some finitary algebra on M? The structures of such closed hulls are related to the disjoint-cycle decompositions of the given elements. If the closed hull is not just the cyclic subgroup on the given element then it is nonminimal as a closed hull and is represented as a subdirect product of finite cyclic groups as well as by a quotient group of a group of infinite sequences. We determine the conditions under which it has a nontrivial primary component for a given prime p and show that such components must be bounded abelian groups.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-1973-0322065-6