The depth of a finite simple group

Burness, Timothy; Liebeck, Martin; Shalev, Aner

doi:10.1090/proc/13937

The depth of a finite simple group
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by Timothy C. Burness, Martin W. Liebeck and Aner Shalev PDF

Proc. Amer. Math. Soc. 146 (2018), 2343-2358 Request permission

Abstract:

We introduce the notion of the depth of a finite group $G$, defined as the minimal length of an unrefinable chain of subgroups from $G$ to the trivial subgroup. In this paper we investigate the depth of (non-abelian) finite simple groups. We determine the simple groups of minimal depth, and show, somewhat surprisingly, that alternating groups have bounded depth. We also establish general upper bounds on the depth of simple groups of Lie type, and study the relation between the depth and the much studied notion of the length of simple groups. The proofs of our main theorems depend (among other tools) on a deep number-theoretic result, namely, Helfgott’s recent solution of the ternary Goldbach conjecture.

References

K. Alladi, R. Solomon, and A. Turull, Finite simple groups of bounded subgroup chain length, J. Algebra 231 (2000), no. 1, 374–386. MR 1779605, DOI 10.1006/jabr.2000.8371
László Babai, On the length of subgroup chains in the symmetric group, Comm. Algebra 14 (1986), no. 9, 1729–1736. MR 860123, DOI 10.1080/00927878608823393
John N. Bray, Derek F. Holt, and Colva M. Roney-Dougal, The maximal subgroups of the low-dimensional finite classical groups, London Mathematical Society Lecture Note Series, vol. 407, Cambridge University Press, Cambridge, 2013. With a foreword by Martin Liebeck. MR 3098485, DOI 10.1017/CBO9781139192576
Ben Brewster, Michael B. Ward, and Irene Zimmermann, Finite groups having chain difference one, J. Algebra 160 (1993), no. 1, 179–191. MR 1237083, DOI 10.1006/jabr.1993.1183
N. Burgoyne, R. Griess, and R. Lyons, Maximal subgroups and automorphisms of Chevalley groups, Pacific J. Math. 71 (1977), no. 2, 365–403. MR 444795
Timothy C. Burness, Martin W. Liebeck, and Aner Shalev, Generation and random generation: from simple groups to maximal subgroups, Adv. Math. 248 (2013), 59–95. MR 3107507, DOI 10.1016/j.aim.2013.07.009
Timothy C. Burness, Martin W. Liebeck, and Aner Shalev, Generation of second maximal subgroups and the existence of special primes, Forum Math. Sigma 5 (2017), Paper No. e25, 41. MR 3720785, DOI 10.1017/fms.2017.21
Peter J. Cameron, Ron Solomon, and Alexandre Turull, Chains of subgroups in symmetric groups, J. Algebra 127 (1989), no. 2, 340–352. MR 1028457, DOI 10.1016/0021-8693(89)90256-1
Arjeh M. Cohen, Martin W. Liebeck, Jan Saxl, and Gary M. Seitz, The local maximal subgroups of exceptional groups of Lie type, finite and algebraic, Proc. London Math. Soc. (3) 64 (1992), no. 1, 21–48. MR 1132853, DOI 10.1112/plms/s3-64.1.21
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
L.E. Dickson, Linear groups with an exposition of the Galois field theory, Teubner, Leipzig 1901 (Dover reprint 1958).
Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
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
Koichiro Harada, Finite simple groups with short chains of subgroups, J. Math. Soc. Japan 20 (1968), 655–672. MR 230811, DOI 10.2969/jmsj/02040655
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
Margaret A. Hartenstein and Ronald M. Solomon, Finite groups of chain difference one, J. Algebra 229 (2000), no. 2, 601–622. MR 1769290, DOI 10.1006/jabr.1999.8262
H.A. Helfgott, The ternary Goldbach problem, Annals of Math. Studies, to appear (see arXiv:1501.05438).
Kenkiti Iwasawa, Über die endlichen Gruppen und die Verbände ihrer Untergruppen, J. Fac. Sci. Imp. Univ. Tokyo Sect. I. 4 (1941), 171–199 (German). MR 0005721
Zvonimir Janko, Finite simple groups with short chains of subgroups, Math. Z. 84 (1964), 428–437. MR 165009, DOI 10.1007/BF01109910
Zvonimir Janko, Finite groups with invariant fourth maximal subgroups, Math. Z. 82 (1963), 82-89, DOI 10.1007/BF01112825. MR0153731
Peter Kleidman and Martin Liebeck, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990. MR 1057341, DOI 10.1017/CBO9780511629235
Martin W. Liebeck, Cheryl E. Praeger, and Jan Saxl, A classification of the maximal subgroups of the finite alternating and symmetric groups, J. Algebra 111 (1987), no. 2, 365–383. MR 916173, DOI 10.1016/0021-8693(87)90223-7
Martin W. Liebeck and Jan Saxl, On point stabilizers in primitive permutation groups, Comm. Algebra 19 (1991), no. 10, 2777–2786. MR 1129540, DOI 10.1080/00927879108824292
Martin W. Liebeck, Jan Saxl, and Gary M. Seitz, Subgroups of maximal rank in finite exceptional groups of Lie type, Proc. London Math. Soc. (3) 65 (1992), no. 2, 297–325. MR 1168190, DOI 10.1112/plms/s3-65.2.297
Leo Murata and Carl Pomerance, On the largest prime factor of a Mersenne number, Number theory, CRM Proc. Lecture Notes, vol. 36, Amer. Math. Soc., Providence, RI, 2004, pp. 209–218. MR 2076597, DOI 10.1090/crmp/036/16
Joseph Petrillo, On the length of finite simple groups having chain difference one, Arch. Math. (Basel) 88 (2007), no. 4, 297–303. MR 2311836, DOI 10.1007/s00013-006-1983-4
Gary M. Seitz, Ron Solomon, and Alexandre Turull, Chains of subgroups in groups of Lie type. II, J. London Math. Soc. (2) 42 (1990), no. 1, 93–100. MR 1078177, DOI 10.1112/jlms/s2-42.1.93
John Shareshian and Russ Woodroofe, A new subgroup lattice characterization of finite solvable groups, J. Algebra 351 (2012), 448–458. MR 2862218, DOI 10.1016/j.jalgebra.2011.10.032
Ron Solomon and Alexandre Turull, Chains of subgroups in groups of Lie type $\textrm {I}$, J. Algebra 132 (1990), no. 1, 174–184. MR 1060841, DOI 10.1016/0021-8693(90)90261-L
Ron Solomon and Alexandre Turull, Chains of subgroups in groups of Lie type. III, J. London Math. Soc. (2) 44 (1991), no. 3, 437–444. MR 1149006, DOI 10.1112/jlms/s2-44.3.437
Michio Suzuki, On a class of doubly transitive groups, Ann. of Math. (2) 75 (1962), 105–145. MR 136646, DOI 10.2307/1970423
I. M. Vinogradov, Representation of an odd number as a sum of three primes, Dokl. Akad. Nauk. SSR (1937), 291–294.