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 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 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 and show that such components must be bounded abelian groups.

**[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.

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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