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A Kurosh subgroup theorem for free pro- $ \mathcal{C}$-products of pro- $ \mathcal{C}$-groups


Authors: Dion Gildenhuys and Luis Ribes
Journal: Trans. Amer. Math. Soc. 186 (1973), 309-329
MSC: Primary 20F20
DOI: https://doi.org/10.1090/S0002-9947-1973-0340433-3
MathSciNet review: 0340433
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathcal{C}$ be a class of finite groups, closed under finite products, subgroups and homomorphic images. In this paper we define and study free pro- $ \mathcal{C}$-products of pro- $ \mathcal{C}$-groups indexed by a pointed topological space. Our main result is a structure theorem for open subgroups of such free products along the lines of the Kurosh subgroup theorem for abstract groups. As a consequence we obtain that open subgroups of free pro- $ \mathcal{C}$-groups on a pointed topological space, are free pro- $ \mathcal{C}$-groups on (compact, totally disconnected) pointed topological spaces.


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

  • [1] E. Binz, J. Neukirch and G. H. Wenzel, Free pro- $ \mathcal{C}$-products, Queen's Mathematical preprint No. 1969-17.
  • [2] N. Bourbaki, Eléments de mathématique. Part. 1. Les structures fondamentales de l'analyse. Livre III: Topologie générale. Chap. II, 3rd ed., Actualités Sci. Indust., no. 858, Hermann, Paris, 1961. MR 0004747 (3:55e)
  • [3] D. Gildenhuys and C. K. Lim, Free pro- $ \mathcal{C}$-groups, Math. Z. 125 (1972), 233-254. MR 0310071 (46:9174)
  • [4] K. W. Gruenberg, Residual properties of infinite soluble groups, Proc. London Math. Soc. (3) 7 (1957), 29-62. MR 19, 386. MR 0087652 (19:386a)
  • [5] -, Projective profinite groups, J. London Math. Soc. 42 (1967), 155-165. MR 35 #260. MR 0209362 (35:260)
  • [6] K. Iwasawa, On solvable extensions of algebraic number fields, Ann. of Math. (2) 58 (1953), 548-572. MR 15, 509. MR 0059314 (15:509d)
  • [7] A. G. Kuroš, Lectures on general algebra, Fizmatgiz, Moscow, 1962; English transl., Chelsea, New York, 1963. MR 25 #5097; 28 #1228. MR 0141700 (25:5097)
  • [8] -, Die Untergruppen der freien Produkte von beliebigen Gruppen, Math. Ann. 109 (1934), 647-660. MR 1512914
  • [9] S. Mac Lane, A proof of the subgroup theorem for free products, Mathematica 5 (1958), 13-19. MR 20 #3911. MR 0097442 (20:3911)
  • [10] W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, Interscience, New York, 1966.
  • [11] J. Neukirch, Freie Produkte pro-endlicher Gruppen und ihre Kohomologie, Arch. Math. (Basel) 22 (1971), 337-357. MR 0347992 (50:490)
  • [12] M. Hall, A topology for free groups and related groups, Ann. of Math. (2) 52 (1950), 127-139. MR 12, 158. MR 0036767 (12:158b)
  • [13] L. Ribes, Introduction to profinite groups and Galois cohomology, Queen's Papers in Pure and Appl. Math., no. 24, Queen's University, Kingston, Ontario, 1970. MR 41 #5495. MR 0260875 (41:5495)
  • [14] -, On amalgamated products of profinite groups, Math. Z. 123 (1971), 357-364. MR 0291295 (45:389)
  • [15] J.-P. Serre, Cohomologie Galoisienne, 3rd ed., Lecture Notes in Math., vol. 5, Springer-Verlag, Berlin and New York, 1965. MR 34 #1328. MR 0201444 (34:1328)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0340433-3
Keywords: Profinite group, pro- $ \mathcal{C}$-group, free product, coproduct, projective limit, pointed topological space, compact, totally disconnected
Article copyright: © Copyright 1973 American Mathematical Society

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