Infinite products of finite simple groups

Saxl, Jan; Shelah, Saharon; Thomas, Simon

doi:10.1090/S0002-9947-96-01746-1

Infinite products of finite simple groups
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by Jan Saxl, Saharon Shelah and Simon Thomas PDF

Trans. Amer. Math. Soc. 348 (1996), 4611-4641 Request permission

Abstract:

We classify the sequences $\langle S_{n} \mid n \in \mathbb {N} \rangle$ of finite simple nonabelian groups such that $\prod _{n} S_{n}$ has uncountable cofinality.

References

Hyman Bass, Some remarks on group actions on trees, Comm. Algebra 4 (1976), no. 12, 1091–1126. MR 419616, DOI 10.1080/00927877608822154
J. L. Brenner, Covering theorems for FINASIGs. VIII. Almost all conjugacy classes in ${\cal A}_{n}$ have exponent $\leq {}4$, J. Austral. Math. Soc. Ser. A 25 (1978), no. 2, 210–214. MR 480718, DOI 10.1017/s1446788700038799
Roger W. Carter, Simple groups of Lie type, Pure and Applied Mathematics, Vol. 28, John Wiley & Sons, London-New York-Sydney, 1972. MR 0407163
Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
Wilfrid Hodges, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993. MR 1221741, DOI 10.1017/CBO9780511551574
T. Jech, Multiple forcing, Cambridge Tracts in Mathematics, vol. 88, Cambridge University Press, Cambridge, 1986. MR 895139
Sabine Koppelberg, Boolean algebras as unions of chains of subalgebras, Algebra Universalis 7 (1977), no. 2, 195–203. MR 434914, DOI 10.1007/BF02485429
S. Koppelberg and J. Tits, Une propriété des produits directs infinis groupes finis isomorphes, C. R. Acad. Sci. Paris Sér. A 279 (1974), 583–585.
Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
H. D. Macpherson and Peter M. Neumann, Subgroups of infinite symmetric groups, J. London Math. Soc. (2) 42 (1990), no. 1, 64–84. MR 1078175, DOI 10.1112/jlms/s2-42.1.64
Stephen P. Humphries, On reducible braids and composite braids, Glasgow Math. J. 36 (1994), no. 2, 197–199. MR 1279892, DOI 10.1017/S0017089500030731
Afzal Beg, On $LC$-, $RC$-, and $C$-loops, Kyungpook Math. J. 20 (1980), no. 2, 211–215. MR 607061
Charles Small, Arithmetic of finite fields, Monographs and Textbooks in Pure and Applied Mathematics, vol. 148, Marcel Dekker, Inc., New York, 1991. MR 1186215
J. D. Sharp and S. Thomas, Uniformisation problems and the cofinality of the infinite symmetric group, Notre Dame Journal of Formal Logic 35 (1994), 328–345.
J. D. Sharp and S. Thomas, Unbounded families and the cofinality of the infinite symmetric group, Arch. Math. Logic 34 (1995), 33–45.
Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955, DOI 10.1007/978-3-662-21543-2
Saharon Shelah, Vive la différence. I. Nonisomorphism of ultrapowers of countable models, Set theory of the continuum (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 26, Springer, New York, 1992, pp. 357–405. MR 1233826, DOI 10.1007/978-1-4613-9754-0_{2}0
Donald E. Taylor, The geometry of the classical groups, Sigma Series in Pure Mathematics, vol. 9, Heldermann Verlag, Berlin, 1992. MR 1189139
S. Thomas, The cofinalities of the infinite dimensional classical groups, J. Algebra 179 (1996), 704–719.
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math 3 (1892), 265–284.