Characterizations of the solvable radical

Flavell, Paul; Guest, Simon; Guralnick, Robert

doi:10.1090/S0002-9939-09-10066-7

Characterizations of the solvable radical
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by Paul Flavell, Simon Guest and Robert Guralnick PDF

Proc. Amer. Math. Soc. 138 (2010), 1161-1170 Request permission

Abstract:

We prove that there exists a constant $k$ with the property: if $\mathcal {C}$ is a conjugacy class of a finite group $G$ such that every $k$ elements of $\mathcal {C}$ generate a solvable subgroup, then $\mathcal {C}$ generates a solvable subgroup. In particular, using the Classification of Finite Simple Groups, we show that we can take $k=4$. We also present proofs that do not use the Classification Theorem. The most direct proof gives a value of $k=10$. By lengthening one of our arguments slightly, we obtain a value of $k=7$.

References

M. Aschbacher, Finite group theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 10, Cambridge University Press, Cambridge, 2000. MR 1777008, DOI 10.1017/CBO9781139175319
Abdullah Al-Roqi and Paul Flavell, On the Fitting height of a soluble group that is generated by a conjugacy class of 3-elements, Bull. Lond. Math. Soc. 39 (2007), no. 6, 973–981. MR 2392820, DOI 10.1112/blms/bdm102
M. Aschbacher and R. Guralnick, Some applications of the first cohomology group, J. Algebra 90 (1984), no. 2, 446–460. MR 760022, DOI 10.1016/0021-8693(84)90183-2
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
Paul Flavell, On the Fitting height of a soluble group that is generated by a conjugacy class, J. London Math. Soc. (2) 66 (2002), no. 1, 101–113. MR 1911223, DOI 10.1112/S0024610702003290
Nikolai Gordeev, Fritz Grunewald, Boris Kunyavskiĭ, and Eugene Plotkin, On the number of conjugates defining the solvable radical of a finite group, C. R. Math. Acad. Sci. Paris 343 (2006), no. 6, 387–392 (English, with English and French summaries). MR 2259878, DOI 10.1016/j.crma.2006.08.005
Nikolai Gordeev, Fritz Grunewald, Boris Kunyavskii, and Eugene Plotkin, A description of Baer-Suzuki type of the solvable radical of a finite group, J. Pure Appl. Algebra 213 (2009), no. 2, 250–258. MR 2467402, DOI 10.1016/j.jpaa.2008.06.006
N. Gordeev, F. Grunewald, B. Kunyavskii, and E. Plotkin, Baer–Suzuki Theorem for the solvable radical of a finite group, preprint.
Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups. Number 3. Part I. Chapter A, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1998. Almost simple $K$-groups. MR 1490581, DOI 10.1090/surv/040.3
Simon Guest, A solvable version of the Baer–Suzuki theorem, Trans. Amer. Math. Soc., to appear.
Robert M. Guralnick and William M. Kantor, Probabilistic generation of finite simple groups, J. Algebra 234 (2000), no. 2, 743–792. Special issue in honor of Helmut Wielandt. MR 1800754, DOI 10.1006/jabr.2000.8357
Robert Guralnick, Eugene Plotkin, and Aner Shalev, Burnside-type problems related to solvability, Internat. J. Algebra Comput. 17 (2007), no. 5-6, 1033–1048. MR 2355682, DOI 10.1142/S0218196707003962
Robert M. Guralnick and Jan Saxl, Generation of finite almost simple groups by conjugates, J. Algebra 268 (2003), no. 2, 519–571. MR 2009321, DOI 10.1016/S0021-8693(03)00182-0
Martin W. Liebeck, The classification of finite simple Moufang loops, Math. Proc. Cambridge Philos. Soc. 102 (1987), no. 1, 33–47. MR 886433, DOI 10.1017/S0305004100067025
Martin W. Liebeck and Jan Saxl, Minimal degrees of primitive permutation groups, with an application to monodromy groups of covers of Riemann surfaces, Proc. London Math. Soc. (3) 63 (1991), no. 2, 266–314. MR 1114511, DOI 10.1112/plms/s3-63.2.266
Gunter Malle, Jan Saxl, and Thomas Weigel, Generation of classical groups, Geom. Dedicata 49 (1994), no. 1, 85–116. MR 1261575, DOI 10.1007/BF01263536
Olaf Manz and Thomas R. Wolf, Representations of solvable groups, London Mathematical Society Lecture Note Series, vol. 185, Cambridge University Press, Cambridge, 1993. MR 1261638, DOI 10.1017/CBO9780511525971