Finite groups with a large automorphism orbit. (English) Zbl 1446.20034
Let \(G\) be a finite group and let \(A=\mathrm{Aut}(G)\). A natural problem is to see to what extent the existence of a large \(A\)-orbit in \(G\) influences the structure of \(G\). The author defines the positive rational number \(\mathrm{maol}(G)\) to be the quotient of the largest size of an \(A\)-orbit in \(G\) and the order of \(G\) and he uses it to establish several quantitative results.
The main results of this long and technical paper (which uses the CFSG) are rather impressive and they are stated in the main Theorem 1.1.2 as follows.
A) If \(\mathrm{maol}(G)\) is larger than \(\frac{18}{19}\), then \(G\) is solvable.
The above value is conceivably not the best one possible and the author is asking the question if \(\mathrm{maol}(G)\leq \frac{3}{2}\) for all nonsolvable \(G\).
B) There exists a function \(f\) defined on \((0, 1]\) and taking positive values such that for every \(x\in (0, 1]\) and each finite \(G\) with \(\mathrm{maol}(G)\geq x\) the orders of the nonabelian composition factors of \(G\) are at most \(f(x)\).
C) For every finite simple nonabelian group \(S\) there exists \(c(S)\in (0, 1]\) such that for each \(N>1\) there exists a finite group \(G\) with \(\mathrm{maol}(G)\geq c(S)\) and such that \(S\) occurs as a composition factor of \(G\) with multiplicity at least \(N\).
The main results of this long and technical paper (which uses the CFSG) are rather impressive and they are stated in the main Theorem 1.1.2 as follows.
A) If \(\mathrm{maol}(G)\) is larger than \(\frac{18}{19}\), then \(G\) is solvable.
The above value is conceivably not the best one possible and the author is asking the question if \(\mathrm{maol}(G)\leq \frac{3}{2}\) for all nonsolvable \(G\).
B) There exists a function \(f\) defined on \((0, 1]\) and taking positive values such that for every \(x\in (0, 1]\) and each finite \(G\) with \(\mathrm{maol}(G)\geq x\) the orders of the nonabelian composition factors of \(G\) are at most \(f(x)\).
C) For every finite simple nonabelian group \(S\) there exists \(c(S)\in (0, 1]\) such that for each \(N>1\) there exists a finite group \(G\) with \(\mathrm{maol}(G)\geq c(S)\) and such that \(S\) occurs as a composition factor of \(G\) with multiplicity at least \(N\).
Reviewer: Marian Deaconescu (Safat)
MSC:
| 20D45 | Automorphisms of abstract finite groups |
| 20D05 | Finite simple groups and their classification |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
References:
| [1] | Abbott, R.; Bray, J.; Linton, S.; Nickerson, S.; Norton, S.; Parker, R.; Suleiman, I.; Tripp, J.; Walsh, P.; Wilson, R., ATLAS of finite group representations - version 3, online resource |
| [2] | Biggs, N. L.; White, A. T., Permutation Groups and Combinatorial Structures, London Mathematical Society Lecture Note Series, vol. 33, (1979), Cambridge University Press: Cambridge University Press Cambridge, (reprinted 2008) · Zbl 0415.05002 |
| [3] | Bors, A., Classification of finite group automorphisms with a large cycle, Comm. Algebra, 44, 11, 4823-4843, (2016) · Zbl 1355.20018 |
| [4] | Bors, A., Finite groups with an automorphism of large order, J. Group Theory, 20, 4, 681-718, (2017) · Zbl 1371.20019 |
| [5] | Cameron, P. J.; Praeger, C. E., Block-transitive t-designs. I. Point-imprimitive designs, Discrete Math., 118, 33-43, (1993) · Zbl 0780.05006 |
| [6] | Cameron, P. J.; Praeger, C. E., Block-transitive t-designs. II. Large t, (De Clerck, F.; etal., Finite Geometry and Combinatorics. Finite Geometry and Combinatorics, London Mathematical Society Lecture Note Series, vol. 191, (1993), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0792.05017 |
| [7] | Cannon, J. J.; Holt, D. F., Computing conjugacy class representatives in permutation groups, J. Algebra, 300, 1, 213-222, (2006) · Zbl 1096.20005 |
| [8] | Dahl, D. A.; Atwood, C. L.; LaViolette, R. A., A random-walk pseudorandom byte generator, Appl. Math. Model., 24, 771-778, (2000) · Zbl 0972.65002 |
| [9] | Dalla Volta, F.; Lucchini, A., Generation of almost simple groups, J. Algebra, 178, 1, 194-223, (1995) · Zbl 0839.20021 |
| [10] | Fulman, J.; Guralnick, R., Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements, Trans. Amer. Math. Soc., 364, 6, 3023-3070, (2012) · Zbl 1256.20048 |
| [11] | GAP - groups, algorithms, and programming, version 4.8.6, (2016) |
| [12] | Gorenstein, D.; Lyons, R.; Solomon, R., The classification of the finite simple groups, number 3, (Mathematical Surveys and Monographs, vol. 40, (1998), American Mathematical Society: American Mathematical Society Providence) · Zbl 0890.20012 |
| [13] | Hartley, B., A general Brauer-Fowler theorem and centralizers in locally finite groups, Pacific J. Math., 152, 1, 101-117, (1992) · Zbl 0713.20023 |
| [14] | Hartley, B.; Kuzucuoğlu, M., Centralizers of elements in locally finite simple groups, Proc. Lond. Math. Soc. (3), 62, 2, 301-324, (1991) · Zbl 0682.20020 |
| [15] | Huber, M., Flag-Transitive Steiner Designs, Frontiers in Mathematics, (2009), Birkhäuser: Birkhäuser Basel · Zbl 1179.05002 |
| [16] | James, G.; Kerber, A., The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and Its Applications, (1981), Addison-Wesley: Addison-Wesley Reading · Zbl 0491.20010 |
| [17] | L’Ecuyer, P., Uniform random number generation, Ann. Oper. Res., 53, 77-120, (1994) · Zbl 0843.65004 |
| [18] | Robbins, H., A remark on Stirling’s formula, Amer. Math. Monthly, 62, 1, 26-29, (1955) · Zbl 0068.05404 |
| [19] | Robinson, D. J.S., A Course in the Theory of Groups, Graduate Texts in Mathematics, vol. 80, (1996), Springer: Springer New York |
| [20] | Rose, J. S., Automorphism groups of groups with trivial centre, Proc. Lond. Math. Soc. (3), 31, 2, 167-193, (1975) · Zbl 0315.20021 |
| [21] | Thas, K., Finite flag-transitive projective planes: a survey and some remarks, Discrete Math., 266, 417-429, (2003) · Zbl 1026.51005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
