Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] L. E. Dickson, History of the theory of numbers. Vol. II, Carnegie Inst., Washington, D. C., 1920.
  • [2] M. Gould, Automorphism groups of algebras of finite type, Canad. J. Math. 24 (1972), 1065-1069. MR 0311547 (47:109)
  • [3] B. Jónsson, Algebraic structures with prescribed automorphism groups, Colloq. Math. 19 (1968), 1-4. MR 36 #6336. MR 0223288 (36:6336)
  • [4] -, Topics in universal algebra, Lecture Notes in Math., vol. 250, Springer-Verlag, Berlin and New York, 1972. MR 0345895 (49:10625)
  • [5] A. Karrass and D. Solitar, Some remarks on the infinite symmetric group, Math. Z. 66 (1956), 64-69. MR 18, 376. MR 0081274 (18:376a)
  • [6] T.-J. Stieltjes, Essai sur la théorie des nombres; premiers élements, Paris, 1895.
  • [7] H. Wielandt, Unendliche Permutationsgruppen, Zweite Vervielfältigung, York University, 1967.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20E99

Retrieve articles in all journals with MSC: 20E99


Additional Information

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

American Mathematical Society