We improve on recent estimates for the probability of generating the
alternating and symmetric groups $\mathrm{Alt}(n)$ and $\mathrm{Sym}(n)$. In
particular we find the sharp lower bound, if the probability is given by a
quadratic in $n^{-1}$. This leads to improved bounds on the largest number
$h(\mathrm{Alt}(n))$ such that a direct product of $h(\mathrm{Alt}(n))$ copies
of $\mathrm{Alt}(n)$ can be generated by two elements

Morgan, Luke

Roney-Dougal, Colva M.

English

arXiv

We improve on recent estimates for the probability of generating the alternating and symmetric groups An and Sn. In particular, we find the sharp lower bound if the probability is given by a quadratic in n−1. This leads to improved bounds on the largest number h(An) such that a direct product of h(An) copies of An can be generated by two elements.</p

Roney-Dougal, Colva Mary

University of St. Andrews - Pure

A note on the probability of generating alternating or symmetric groups

Luke Morgan

Colva M. Roney-Dougal

Crossref

Archiv der Mathematik

University of St Andrews Research Portal

The research of the first author is supported by the Australian Research Council grant DP120100446.We improve on recent estimates for the probability of generating the alternating and symmetric groups An and Sn. In particular, we find the sharp lower bound if the probability is given by a quadratic in n−1. This leads to improved bounds on the largest number h(An) such that a direct product of h(An) copies of An can be generated by two elements.Peer reviewe

St Andrews Research Repository

arXiv:submit/1294667  [math.GR]  2 Jul 2015A NOTE ON THE PROBABILITY OF GENERATING ALTERNATING ORSYMMETRIC GROUPSLUKE MORGAN AND COLVA M. RONEY-DOUGALAbstract. We improve on recent estimates for the probability of generating the alternating and symmetricgroups An and Sn. In particular we find the sharp lower bound, if the probability is given by a quadratic inn−1. This leads to improved bounds on the largest number h(An) such that a direct product of h(An) copiesof An can be generated by two elements.1. IntroductionFor a group X = Sn or An, we write p(X) for the probability that two elements of X generate a group thatcontains An. In [1], Dixon proved that p(Sn) → 1 as n→∞. In [2] he sharpened this statement top(Sn) = 1−1n−1n2−4n3−23n4−171n5−1542n6+O(n−7).For many applications, numerical results are needed, rather than asymptotics. In [5] Maro´ti and Tamburiniproved explicit upper and lower bounds1−1n−13n2< p(X) 6 1−1n+23n2.In this present note, we find the best possible lower bound of this type, and a close-to-optimal upper bound.Theorem 1.1. Let X = An or X = Sn with n > 5. Then1−1n−8.8n26 p(X) < 1−1n−0.93n2.Equality holds in the lower bound if and only if n = 6.In fact, for n > 14, we prove that 1− 1n− 7.5n2< p(X) < 1− 1n− 0.93n2. The result for smaller n comes fromthe values for p(X) in Table 1 (taken from [7, Table 4.1]).Hall [3] considered the largest number h(S) such that a direct product of h(S) copies of a non-abelian finitesimple group S can be generated by two elements, and proved that h(S) = p(S)|S|/|Out(S)|. The functionh(S) has received considerable attention recently; we refer the reader to [5] for more discussion and referencesand to [6] for lower bounds on h(S) for all non-abelian finite simple groups S. The new bounds above yield:Corollary 1.2. Let n be an integer with n > 14. Then(1−1n−7.5n2)(n!4)< h(An) <(1−1n−0.93n2)(n!4).Let m(S) denote the minimal index of a proper subgroup of a group S. In [4], it is proved that there existabsolute constants c1 and c2 such that 1 − c1/m(S) < p(S) < 1 − c2/m(S), for all non-abelian finite simplegroups S. Our main theorem immediately yields the following bounds on c1 and c2.Corollary 1.3. For n > 5,1−2.468n< p(An) < 1−1.186n,and hence c1 > 2.468 and c2 6 1.186.1991 Mathematics Subject Classification. 20B30; 20P05.Key words and phrases. Symmetric group; alternating group; generation; probability.The research of the first author is supported by the Australian Research Council grant DP120100446. This work was donewhilst the second author was visiting The University of Western Australia as a Cheryl E. Praeger Visiting Research Fellow. Theauthors would like to thank the anonymous referee for their extremely helpful remarks.12 LUKE MORGAN AND COLVA M. RONEY-DOUGAL2. Proof of Theorem 1.1Definition 2.1. For X = An or Sn we let pintrans(X) and ptrans(X) be the probability that two elementschosen randomly from X generate a subgroup of an intransitive maximal subgroup of X , or a subgroup of atransitive maximal subgroup of X other than An, respectively.Lemma 2.2. Let X = An or Sn with n > 14. Thenpintrans(X) <1n+2.7n2.Proof. We prove the result for Sn, the arguments for An are identical. Let x, y ∈ Sn and suppose that Y := 〈x, y〉is contained in an intransitive maximal subgroup. Then Y is contained in a subgroup conjugate to Sk × Sn−kfor some 1 6 k 6 ⌊n−12 ⌋.Let k ∈ {1, . . . , n− 1}. Then the probability that Y 6 Sk × Sn−k is bounded by(nk)(k!(n− k)!n!)2=(nk)−1.So the probability that Y 6 S1×Sn−1 is at most1n, and the probability that Y 6 S2×Sn−2 and Y is transitiveon the orbit of size 2 is bounded by342n(n− 1)=32n(n− 1).Similarly, the probability that Y 6 S3 × Sn−3 and Y is transitive on the orbit of length 3 is1318(n3)−1=133n(n− 1)(n− 2).Now the probability that Y 6 Sk × Sn−k for some 4 6 k 6 ⌊n−12 ⌋ is⌊n−12⌋∑k=41(nk) 6⌊n−12⌋∑k=41(n4) 6 12(n− 7)n(n− 1)(n− 2)(n− 3).We now observe that, since n > 14,32n(n− 1)+133n(n− 1)(n− 2)+12(n− 7)n(n− 1)(n− 2)(n− 3)<2.7n2which completes the proof. Lemma 2.3. Let X = An or Sn, with n > 14. Thenpintrans(X) >1n+0.93n2.Proof. We observe that pintrans(X) is bounded below by the probability that a random pair of elements of Xgenerate a subgroup with a fixed point, or with an orbit of size 2. For X = Sn, we bound pintrans(X) by doinginclusion-exclusion to depth 2 on the union of the sets (Sn)α, with 1 6 α 6 n, and (Sn){α,β} \ (Sn)(α,β), with1 6 α < β 6 n. We find that pintrans(X) is greater than1n+ 342(n−2)!n! −(n−2)!2n! −34(n1)(n−12) (2(n−3)!n!)2−(34)2 (n2)(n−22)2(4(n−4)!n!)2Thuspintrans(X) >1n+8n2 − 52n+ 758n(n− 1)(n− 2)(n− 3)which, since n > 14, is greater than 1n+ 0.93n2. Proof of Theorem 1.1. For the upper bound we use Lemma 2.3. For the lower bound, note that1− p(X) = pintrans(X) + ptrans(X).It follows from the proofs of [5, Lemmas 3.1 and 4.3] that ptrans(X) 64.8n2. Combining this with Lemma 2.2gives the theorem. In Table 1 we record the value of p(An) and p(Sn) for n 6 13, together with our lower and upper bounds asstated in Theorem 1.1. All values are correct to three decimal places.GENERATING ALTERNATING OR SYMMETRIC GROUPS 3Table 1. Precise values and bounds for p(X)n 5 6 7 8 9 10 11 12 13p(An) = 0.633 0.588 0.726 0.739 0.848 0.875 0.893 0.902 0.913p(Sn) = 0.633 0.588 0.795 0.796 0.859 0.875 0.894 0.903 0.913p(X) > 0.448 0.588 0.677 0.737 0.780 0.812 0.836 0.855 0.871p(X) 6 0.763 0.808 0.839 0.861 0.878 0.891 0.902 0.911 0.918References[1] J. D. Dixon. The probability of generating the symmetric group. Math. Z 110 (1969) 199-205.[2] J. D. Dixon. Asymptotics of generating the symmetric and alternating groups. Electron. J. Combin. 12 (2005), Research paper56, 5 pp.[3] P. Hall. The Eulerian function of a group. Quart. J. Math. Oxford 7 (1936), 133–141.[4] M.W. Liebeck & A. Shalev. Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotszky. J. Algebra 184(1996) 31–57.[5] A. Maro´ti & M.C. Tamburini. Bounds for the probability of generating the symmetric and alternating groups.Arch. Math. (Basel) 96(2) (2011) 115–121.[6] N.E. Menezes, M. Quick & C.M. Roney-Dougal. The probability of generating a finite simple group. Israel J. Math 198 (2013)371–392.[7] N.E. Menezes. Random generation and chief length of finite groups. PhD thesis, University of St Andrews (2013).Luke Morgan, School of Mathematics and Statistics,, University of Western Australia,, 35 Stirling Highway,,Crawley, WA 6009, AustraliaE-mail address: luke.morgan@uwa.edu.auColva M. Roney-Dougal, School of Mathematics and Statistics,, University of St Andrews,, North Haugh,, StAndrews,, Fife KY16 9SS, U. K.E-mail address: colva.roney-dougal@st-andrews.ac.uk

https://doi.org/10.1007/s00013-015-0796-8

A note on the probability of generating alternating or symmetric groups

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