Probabilistic generation of finite almost simple groups.(English)Zbl 1548.20107
A finite group \(G\) is said to be \(\frac{3}{2}\)-generated if every non-trivial element belongs to a generating pair.T. Breuer et al., J. Algebra 320, No. 2, 443–494 (2008;Zbl 1181.20013) proved that \(G\) is \(\frac{3}{2}\)-generated if and only if every proper quotient of \(G\) is cyclic.W. M. Kantor andA. Lubotzky [Geom. Dedicata 36, No. 1, 67–87 (1990;Zbl 0718.20011)] asked whether there was a probabilistic version of \(\frac{3}{2}\)-generation. This is not the case for alternating groups: if \(x \in A_{n}\) moves only a bounded number of points, the probability that \(x\) and a random element of \(A_{n}\) generate a transitive group goes to 0 as \(n \rightarrow \infty\).
The main result of the paper under review is Theorem 1.1: There exists an absolute constant \(\varepsilon > 0\) such that the following holds. Let \(S\) be a finite simple group of Lie type of large enough order and let \(x,y \in \operatorname{Aut}(S)\) with \(x \not =1\). Then the probability that \(x\) and a random element of \(Sy\) generate \(\langle S,x,y \rangle\) is at least \(\varepsilon\).
As a consequence of the main theorem, the authors prove Corollary 1.3: Let \(G\) be a profinite group and let \(g\in G\). Then, the following are equivalent: (i) the probability that \(g\) and a random element of \(G\) generate a prosolvable group is positive; (ii) there exists \(C \geq 1\) such that \(g\) centralizes all but at most \(C\) non-abelian chief factors of \(G/N\) for every open normal subgroup \(N\) of \(G\).
A key ingredient in the proof of Theorem 1.1 is a result of independent interest. The authors consider the proportion of elements in a classical group of dimension \(n\) over the field of size \(q\) which fix no subspace of dimension at most \(t\). For \(q\) and \(t\) fixed, they prove that the limit as \(n \rightarrow \infty\) exists and is strictly between \(0\) and \(1\).
The main result of the paper under review is Theorem 1.1: There exists an absolute constant \(\varepsilon > 0\) such that the following holds. Let \(S\) be a finite simple group of Lie type of large enough order and let \(x,y \in \operatorname{Aut}(S)\) with \(x \not =1\). Then the probability that \(x\) and a random element of \(Sy\) generate \(\langle S,x,y \rangle\) is at least \(\varepsilon\).
As a consequence of the main theorem, the authors prove Corollary 1.3: Let \(G\) be a profinite group and let \(g\in G\). Then, the following are equivalent: (i) the probability that \(g\) and a random element of \(G\) generate a prosolvable group is positive; (ii) there exists \(C \geq 1\) such that \(g\) centralizes all but at most \(C\) non-abelian chief factors of \(G/N\) for every open normal subgroup \(N\) of \(G\).
A key ingredient in the proof of Theorem 1.1 is a result of independent interest. The authors consider the proportion of elements in a classical group of dimension \(n\) over the field of size \(q\) which fix no subspace of dimension at most \(t\). For \(q\) and \(t\) fixed, they prove that the limit as \(n \rightarrow \infty\) exists and is strictly between \(0\) and \(1\).
Reviewer: Enrico Jabara (Venezia)
MSC:
| 20P05 | Probabilistic methods in group theory |
| 20D06 | Simple groups: alternating groups and groups of Lie type |
| 20F69 | Asymptotic properties of groups |
| 20G40 | Linear algebraic groups over finite fields |
Keywords:
generation of almost simple groups;probabilistic generation;spread;\( \frac{3}{2}\)-generation;random elementCite
References:
| [1] | Breuer, T.; Guralnick, R. M.; Kantor, W. M., Probabilistic generation of finite simple groups, II, J. Algebra, 320, 443-494, 2008 ·Zbl 1181.20013 |
| [2] | Britnell, J., Cyclic, separable and semisimple matrices in the special linear groups over a finite field, J. Lond. Math. Soc., 66, 605-622, 2002 ·Zbl 1050.20034 |
| [3] | Britnell, J., Cyclic, separable and semisimple transformations in the special unitary groups over a finite field, J. Group Theory, 9, 547-569, 2006 ·Zbl 1105.20041 |
| [4] | Burness, T. C.; Guralnick, R. M.; Harper, S., The spread of a finite group, Ann. Math., 193, 619-687, 2021 ·Zbl 1480.20081 |
| [5] | T.C. Burness, R.M. Guralnick, S. Harper, Probabilistic \(\frac{ 3}{ 2} \)-generation of finite groups, preprint. ·Zbl 1480.20081 |
| [6] | Dalla Volta, F.; Lucchini, A., Generation of almost simple groups, J. Algebra, 2, 194-223, 1995 ·Zbl 0839.20021 |
| [7] | Dixon, J. D., The probability of generating the symmetric group, Math. Z., 110, 199-205, 1969 ·Zbl 0176.29901 |
| [8] | Eberhard, S.; Garzoni, D., Conjugacy classes of derangements in finite groups of Lie type, Trans. Am. Math. Soc., 2024, in press |
| [9] | Fulman, J., Cycle indices for the finite classical groups, J. Group Theory, 2, 251-289, 1999 ·Zbl 0943.20048 |
| [10] | Fulman, J.; Guralnick, R. M., Conjugacy class properties of the extension of \(G L(n, q)\) generated by the inverse transpose involution, J. Algebra, 275, 356-396, 2004 ·Zbl 1065.20065 |
| [11] | Fulman, J.; Guralnick, R. M., Derangements in subspace actions of finite classical groups, Trans. Amer. Math. Soc., 369, 2521-2572, 1999 ·Zbl 1431.20033 |
| [12] | Fulman, J.; Neumann, P. M.; Praeger, C. E., A generating function approach to the enumeration of matrices in classical groups over finite fields, Mem. Am. Math. Soc., 830, 2005 ·Zbl 1082.05097 |
| [13] | Garzoni, D.; McKemmie, E., On the probability of generating invariably a finite simple group, J. Pure Appl. Algebra, 227, 2023 ·Zbl 1514.20056 |
| [14] | Guralnick, R. M.; Kantor, W. M., Probabilistic generation of finite simple groups, J. Algebra, 234, 743-792, 2000 ·Zbl 0973.20012 |
| [15] | Guralnick, R. M.; Kantor, W. M.; Saxl, J., The probability of generating a classical group, Commun. Algebra, 22, 1395-1402, 1994 ·Zbl 0820.20022 |
| [16] | Guralnick, R. M.; Larsen, M.; Tiep, P. H., Representation growth in positive characteristic and conjugacy classes of maximal subgroups, Duke Math. J., 161, 107-137, 2012 ·Zbl 1244.20007 |
| [17] | Kantor, W. M.; Lubotzky, A., The probability of generating a finite classical group, Geom. Dedic., 36, 67-87, 1990 ·Zbl 0718.20011 |
| [18] | Kleidman, P. B.; Liebeck, M. W., The Subgroup Structure of the Finite Classical Groups, vol. 129, 1990, Cambridge University Press ·Zbl 0697.20004 |
| [19] | Liebeck, M. W.; Saxl, J., Minimal degrees of primitive permutation groups, with an application to monodromy groups of covers of Riemann surfaces, Proc. Lond. Math. Soc., 3, 266-314, 1991 ·Zbl 0696.20004 |
| [20] | Liebeck, M. W.; Shalev, A., The probability of generating a finite simple group, Geom. Dedic., 56, 103-113, 1995 ·Zbl 0836.20068 |
| [21] | Liebeck, M. W.; Shalev, A., Simple groups, permutation groups, and probability, J. Amer. Math. Soc., 12, 497-520, 1999 ·Zbl 0916.20003 |
| [22] | Lucchini, A., Solubilizers in profinite groups, J. Algebra, 647, 619-632, 2024 ·Zbl 1536.20040 |
| [23] | Neumann, P. M.; Praeger, C. E., Derangements and eigenvalue-free elements in finite classical groups, J. Lond. Math. Soc., 58, 564-586, 1998 ·Zbl 0936.15020 |
| [24] | Steinberg, R., Generators for simple groups, Can. J. Math., 14, 277-283, 1962 ·Zbl 0103.26204 |
| [25] | Stong, R., Some asymptotic results on finite vector spaces, Adv. Appl. Math., 9, 167-199, 1988 ·Zbl 0681.05004 |
| [26] | Wall, G. E., On the conjugacy classes in the unitary, symplectic, and orthogonal groups, J. Aust. Math. Soc., 3, 1-63, 1963 ·Zbl 0122.28102 |
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