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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Closed hulls in infinite symmetric groups


Author: Franklin Haimo
Journal: Trans. Amer. Math. Soc. 180 (1973), 475-484
MSC: Primary 20E99
DOI: https://doi.org/10.1090/S0002-9947-1973-0322065-6
MathSciNet review: 0322065
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Abstract: Let $ \operatorname{Sym} M$ be the symmetric group of an infinite set $ M$. What is the smallest subgroup of $ \operatorname{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.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0322065-6
Keywords: Closed hull, infinite symmetric group, finitary algebra, disjoint-cycle decomposition, subdirect product, primary component, Yih-hing (Stieltjes) theorem
Article copyright: © Copyright 1973 American Mathematical Society