## Closed hulls in infinite symmetric groups

HTML articles powered by AMS MathViewer

- by Franklin Haimo
- Trans. Amer. Math. Soc.
**180**(1973), 475-484 - DOI: https://doi.org/10.1090/S0002-9947-1973-0322065-6
- PDF | 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.## References

- L. E. Dickson,
- Matthew Gould,
*Automorphism groups of algebras of finite type*, Canadian J. Math.**24**(1972), 1065–1069. MR**311547**, DOI 10.4153/CJM-1972-109-0 - Bjarni Jónsson,
*Algebraic structures with prescribed automorphism groups*, Colloq. Math.**19**(1968), 1–4. MR**223288**, DOI 10.4064/cm-19-1-1-4 - Bjarni Jónsson,
*Topics in universal algebra*, Lecture Notes in Mathematics, Vol. 250, Springer-Verlag, Berlin-New York, 1972. MR**0345895**, DOI 10.1007/BFb0058648 - A. Karrass and D. Solitar,
*Some remarks on the infinite symmetric groups*, Math. Z.**66**(1956), 64–69. MR**81274**, DOI 10.1007/BF01186596
T.-J. Stieltjes,

*History of the theory of numbers*. Vol. II, Carnegie Inst., Washington, D. C., 1920.

*Essai sur la théorie des nombres; premiers élements*, Paris, 1895. H. Wielandt,

*Unendliche Permutationsgruppen*, Zweite Vervielfältigung, York University, 1967.

## Bibliographic 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