Abstract
The first part of the paper establishes results about products of commutators in a d-generator finite group G, for example: if H⊲G=〈g 1,…,g r 〉 then every element of the subgroup [H,G] is a product of f(r) factors of the form \([h_{1},g_{1}][h_{1}^{\prime},g_{1}^{-1}]\ldots\lbrack h_{r},g_{r}][h_{r}^{\prime },g_{r}^{-1}]\) with \(h_{1},h_{1}^{\prime},\ldots,\allowbreak h_{r},h_{r}^{\prime }\in H\). Under certain conditions on H, a similar conclusion holds with the significantly weaker hypothesis that G=H〈g 1,…,g r 〉, where f(r) is replaced by f 1(d,r). The results are applied in the second part of the paper to the study of normal subgroups in finitely generated profinite groups, and in more general compact groups. Results include the characterization of (topologically) finitely generated compact groups which have a countably infinite image, and of those which have a virtually dense normal subgroup of infinite index. As a corollary it is deduced that a compact group cannot have a finitely generated infinite abstract quotient.
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References
Aschbacher, M., Guralnick, R.M.: Some applications of the first cohomology group. J. Algebra 90, 446–460 (1984)
Aschbacher, M.: Finite Group Theory. Cambridge Univ. Press, Cambridge (1988)
Blau, H.: A fixed-point theorem for central elements in quasisimple groups. Proc. Am. Math. Soc. 122, 79–84 (1994)
Babai, L., Cameron, P.J., Pálfy, P.: On the orders of primitive groups with restricted non-abelian composition factors. J. Algebra 79, 161–168 (1982)
Babai, L., Nikolov, N., Pyber, L.: Product growth and mixing in finite groups. In: 19th ACM-SIAM Symposium on Discrete Algorithms, pp. 248–257. SIAM, Philadelphia (2008)
Bump, D.: Lie Groups. Springer, New York (2004)
Carter, R.W.: Finite Groups of Lie Type: Conjugacy Classes and Complex Characters. Wiley and Sons, London (1985)
Dixon, J.D., Mortimer, B.: Permutation Groups. Springer, New York (1996)
Fulman, J., Guralnick, R.: Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements. arXiv:0902.2238 [abs]
Frayne, T., Morel, A., Scott, D.: Reduced direct products. Fundam. Math. 51, 195–228 (1962)
Fried, M., Jarden, M.: Field Arithmetic. Springer, Berlin (1986)
Garion, S., Shalev, A.: Commutator maps, measure preservation, and T-systems. Trans. Am. Math. Soc. 361, 4631–4651 (2009)
Gaschütz, W.: Zu einem von B. H. und H. Neumann gestellten Problem. Math. Nachr. 14, 249–252 (1955)
Guralnick, R.M., Lübeck, F.: On p-singular elements in Chevalley groups in characteristic p. In: Groups and Computation III. Ohio State Univ. Math. Res. Inst. Publ., vol. 8, pp. 169–182. de Gruyter, Berlin (2001)
Gorenstein, D., Lyons, R., Solomon, R.: The Classification of the Finite Simple Groups, No. 3. American Math. Soc., Providence (1998)
Gluck, D., Seress, A., Shalev, A.: Bases for primitive permutation groups and a conjecture of Babai. J. Algebra 199, 367–378 (1998)
Hofmann, K.H., Morris, S.A.: The Structure of Compact Groups, 2nd edn. de Gruyter Studies in Mathematics, vol. 25. Walter de Gruyter, Berlin (2006)
Jones, G.A.: Varieties and simple groups. J. Aust. Math. Soc. 17, 163–173 (1974)
Jaikin-Zapirain, A.: On linear just infinite pro-p groups. J. Algebra 255, 392–404 (2002)
Kleidman, P., Liebeck, M.: The Subgroup Structure of the Finite Classical Groups. LMS Lect. Notes, vol. 129. Cambridge Univ. Press, Cambridge (1990)
Kapovich, M., Leeb, B.: On asymptotic cones and quasi-isometry of fundamental groups of 3-manifolds. Geom. Anal. Funct. Anal. 5, 582–603 (1995)
Landazuri, V., Seitz, G.M.: On the minimal degrees of projective representations of the finite Chevalley groups. J. Algebra 32, 418–443 (1974)
Liebeck, M.W., Shalev, A.: Diameters of finite simple groups: sharp bounds and applications. Ann. Math. 154, 383–406 (2001)
Liebeck, M.W., Shalev, A.: Fuchsian groups, finite simple groups and representation varieties. Invent. Math. 159, 317–367 (2005)
Liebeck, M., O’Brien, E., Shalev, A., Tiep, P.: The Ore conjecture. J. Eur. Math. Soc. 12, 939–1008 (2010)
Liebeck, M., O’Brien, E., Shalev, A., Tiep, P.: Commutators in finite quasisimple groups. Bull. Lond. Math. Soc. 43, 1079–1092 (2011)
Martinez, C., Zelmanov, E.: Products of powers in finite simple groups. Isr. J. Math. 96, 469–479 (1996)
Nikolov, N., Segal, D.: On finitely generated profinite groups, I: strong completeness and uniform bounds. Ann. Math. 165, 171–238 (2007)
Nikolov, N., Segal, D.: On finitely generated profinite groups, II: products in quasisimple groups. Ann. Math. 165, 239–273 (2007)
Nikolov, N., Segal, D.: Powers in finite groups. Groups Geom. Dyn. 5, 501–507 (2011)
Saxl, J., Wilson, J.S.: A note on powers in simple groups. Math. Proc. Camb. Philos. Soc. 122, 91–94 (1997)
Segal, D.: Closed subgroups of profinite groups. Proc. Lond. Math. Soc. 81, 29–54 (2000)
Segal, D.: Words: Notes on Verbal Width in Groups. London Math. Soc. Lecture Notes Series, vol. 361. Cambridge Univ. Press, Cambridge (2009)
Serre, J.-P.: Topics in Galois Theory. Res. Notes Math., vol. 1. Jones and Bartlett, Boston (1992)
Wilson, J.S.: On simple pseudofinite groups. J. Lond. Math. Soc. 51, 471–490 (1995)
Zelmanov, E.I.: On the restricted Burnside problem. In: Proc. Internl. Congress Math. Kyoto 1990, pp. 395–402. Math. Soc. Japan, Tokyo (1991)
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Nikolov, N., Segal, D. Generators and commutators in finite groups; abstract quotients of compact groups. Invent. math. 190, 513–602 (2012). https://doi.org/10.1007/s00222-012-0383-6
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DOI: https://doi.org/10.1007/s00222-012-0383-6