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Transactions of the American Mathematical Society

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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
MathSciNet review: 0340433
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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.

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  • [1] E. Binz, J. Neukirch and G. H. Wenzel, Free pro- $ \mathcal{C}$-products, Queen's Mathematical preprint No. 1969-17.
  • [2] N. Bourbaki, Éléments de mathématique. Part I. Les structures fondamentales de l’analyse. Livre III. Topologie générale. Chapitres I et II, Actual. Sci. Ind., no. 858, Hermann & Cie., Paris, 1940 (French). MR 0004747
  • [3] Dion Gildenhuys and Chong Keang Lim, Free pro-𝒞-groups, Math. Z. 125 (1972), 233–254. MR 0310071
  • [4] K. W. Gruenberg, Residual properties of infinite soluble groups, Proc. London Math. Soc. (3) 7 (1957), 29–62. MR 0087652
  • [5] K. W. Gruenberg, Projective profinite groups, J. London Math. Soc. 42 (1967), 155–165. MR 0209362
  • [6] Kenkichi Iwasawa, On solvable extensions of algebraic number fields, Ann. of Math (2) 58 (1953), 548–572. MR 0059314
  • [7] A. G. Kuroš, Lektsii po obshchei algebre, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1962 (Russian). MR 0141700
  • [8] Alexander Kurosch, Die Untergruppen der freien Produkte von beliebigen Gruppen, Math. Ann. 109 (1934), no. 1, 647–660 (German). MR 1512914, 10.1007/BF01449159
  • [9] Saunders Mac Lane, A proof of the subgroup theorem for free products, Mathematika 5 (1958), 13–19. MR 0097442
  • [10] W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, Interscience, New York, 1966.
  • [11] Jürgen Neukirch, Freie Produkte pro-endlicher Gruppen und ihre Kohomologie, Arch. Math. (Basel) 22 (1971), 337–357 (German). MR 0347992
  • [12] Marshall Hall Jr., A topology for free groups and related groups, Ann. of Math. (2) 52 (1950), 127–139. MR 0036767
  • [13] Luis Ribes, Introduction to profinite groups and Galois cohomology, Queen’s Papers in Pure and Applied Mathematics, No. 24, Queen’s University, Kingston, Ont., 1970. MR 0260875
  • [14] Luis Ribes, On amalgamated products of profinite groups, Math. Z. 123 (1971), 357–364. MR 0291295
  • [15] Jean-Pierre Serre, Cohomologie galoisienne, With a contribution by Jean-Louis Verdier. Lecture Notes in Mathematics, No. 5. Troisième édition, vol. 1965, Springer-Verlag, Berlin-New York, 1965 (French). MR 0201444

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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