Closed hulls in infinite symmetric groups
Author:
Franklin Haimo
Journal:
Trans. Amer. Math. Soc. 180 (1973), 475484
MSC:
Primary 20E99
MathSciNet review:
0322065
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Abstract: Let be the symmetric group of an infinite set . What is the smallest subgroup of 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 ? The structures of such closed hulls are related to the disjointcycle 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 and show that such components must be bounded abelian groups.
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 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), 10651069. MR 0311547 (47:109)
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 B. Jónsson, Algebraic structures with prescribed automorphism groups, Colloq. Math. 19 (1968), 14. MR 36 #6336. MR 0223288 (36:6336)
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 , Topics in universal algebra, Lecture Notes in Math., vol. 250, SpringerVerlag, 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), 6469. 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.
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DOI:
http://dx.doi.org/10.1090/S00029947197303220656
PII:
S 00029947(1973)03220656
Keywords:
Closed hull,
infinite symmetric group,
finitary algebra,
disjointcycle decomposition,
subdirect product,
primary component,
Yihhing (Stieltjes) theorem
Article copyright:
© Copyright 1973
American Mathematical Society
