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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Closed hulls in infinite symmetric groups
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by Franklin Haimo PDF
Trans. Amer. Math. Soc. 180 (1973), 475-484 Request permission

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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Additional Information
  • © Copyright 1973 American Mathematical Society
  • 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