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On a relation between the rank and the proportion of derangements in finite transitive permutation groups
Journal of Combinatorial Theory, Series A 136 (2015) 198–200
Contents lists available at ScienceDirect
Journal of Combinatorial Theory, Series A www.elsevier.com/locate/jcta
On a relation between the rank and the proportion of derangements in finite transitive permutation groups R. Guralnick a,1 , I.M. Isaacs b , P. Spiga c a
Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA b University of Wisconsin, Department of Mathematics, 480 Lincoln drive, Madison, WI 53706, USA c University of Milano-Bicocca, Dipartimento di Matematica Pura e Applicata, Via Cozzi 55, 20125 Milano, Italy
a r t i c l e
i n f o
Article history: Received 17 June 2015 Available online xxxx Keywords: Derangements Rank Permutation groups
a b s t r a c t Let G be a finite transitive group of rank r. We give a short proof that the proportion of derangements in G is at most 1 − 1/r and we classify the permutation groups attaining this bound. © 2015 Elsevier Inc. All rights reserved.
Let G be a finite transitive group on the set Ω. An element g ∈ G is called a derangement if g has no fixed points on Ω, that is, αg = α for each α ∈ Ω. Moreover, the rank of G is the number of orbits of G in its natural component-wise action on the Cartesian product Ω × Ω. The group action is said to be regular if the identity is the only element of G fixing an element of Ω.
E-mail addresses: [email protected] (R. Guralnick), [email protected] (I.M. Isaacs), [email protected] (P. Spiga). 1 Guralnick was partially supported by NSF grant DMS-1302886. http://dx.doi.org/10.1016/j.jcta.2015.07.003 0097-3165/© 2015 Elsevier Inc. All rights reserved.
R. Guralnick et al. / Journal of Combinatorial Theory, Series A 136 (2015) 198–200
199
The main result of this paper is the following. Theorem 1. If G is a finite transitive group on Ω of rank r, then the proportion of elements of G which are derangements on Ω is at most 1 − 1/r. Moreover, this upper bound is met if and only if G acts regularly. In light of Theorem 1, one might wonder whether, by excluding the family of regular permutation groups, the upper bound 1 − 1/r can be improved further. This is not the case. For example, in the group PSL2 (q) in its natural 2-transitive action, the proportion 1 1 of derangements is 12 − q+1 when q is odd and 12 − 2(q+1) when q is even, and this number tends to 1/2 as q tends to infinity. In the Suzuki group 2 B2 (q) and in the Ree group 2 G2 (q) in their natural 2-transitive actions on the points of a Steiner system, the proportion of q 2 +1 1 derangements are 21 − 2(qq+1 2 +1) and 2 − q 3 +1 (respectively), and both of these ratios tend to 1/2 as q tends to infinity. Similar examples can be obtained for larger ranks. The proof of Theorem 1 uses some character theory and is very much in the same spirit as the proof of the beautiful Cameron–Cohen lower bound (r − 1)/|Ω| for the proportion of derangements [2]. This lower bound has motivated a conjecture due independently to Shalev and Boston et al. [1], and in turn this conjecture has spurred some interesting research culminating with a complete solution by the first author and Fulman [4,7,6,5]. The first half of Theorem 1 was also proved in [3, Theorem 3.1] using the so-called second moment method, which is a standard result in probability theory. (The proof of [3, Theorem 3.1] is also very short.) Our proof only uses the Cauchy–Schwarz inequality and characterises the groups attaining the bound. Proof of Theorem 1. Let π be the permutation character of G and let D be the set of derangements of G. Observe that π, π = r and π(g) = 0 for every g ∈ D, and hence
|G|r =
π(g)2 =
g∈G
g∈G\D
⎛ ⎞2 1 ⎝ π(g)2 ≥ π(g)⎠ , |G| − |D| g∈G\D
where the last inequality is a direct application of the Cauchy–Schwarz inequality. As π(g) = 0 for every g ∈ D, this yields ⎛ |G|r ≥
1 ⎝ |G| − |D|
g∈G
⎞2 π(g)⎠ =
|G|2 |G| − |D|
and the first half of the theorem follows. Suppose now that |D|/|G| = 1 − 1/r. By chasing the previous series of inequalities, 2 we get g∈G\D π(g)2 = g∈G\D π(g) . In turn, the Cauchy–Schwarz inequality gives π(g) = π(h) for every g, h ∈ G \ D, that is, π(g) = π(1) = |Ω| for every g ∈ G \ D. Since
200
R. Guralnick et al. / Journal of Combinatorial Theory, Series A 136 (2015) 198–200
G acts faithfully, we must have G \ D = {1} and hence the identity of G is the only permutation fixing a point, that is, G is regular. 2 References [1] N. Boston, W. Dabrowski, T. Foguel, et al., The proportion of fixed-point-free elements in a transitive group, Comm. Algebra 21 (1993) 3259–3275. [2] P.J. Cameron, A.M. Cohen, On the number of fixed-point-free elements in a permutation group, Discrete Math. 106 (1992) 135–138. [3] P. Diaconis, J. Fulman, R. Guralnick, On fixed points of permutations, J. Algebraic Combin. 28 (2008) 189–218. [4] J. Fulman, R. Guralnick, Derangements in simple and primitive groups, in: Groups, Combinatorics, and Geometry, Durham, 2001, World Sci. Publ., River Edge, NJ, 2003, pp. 99–121. [5] J. Fulman, R. Guralnick, Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements, Trans. Amer. Math. Soc. 364 (2012) 3023–3070. [6] J. Fulman, R. Guralnick, Derangements in subspace actions of finite classical groups, Trans. Amer. Math. Soc. (2015), in press. [7] J. Fulman, R. Guralnick, Derangements in finite classical groups for actions related to extension field and imprimitive subgroups and the solution of the Boston–Shalev conjecture, preprint.
Contents lists available at ScienceDirect
Journal of Combinatorial Theory, Series A www.elsevier.com/locate/jcta
On a relation between the rank and the proportion of derangements in finite transitive permutation groups R. Guralnick a,1 , I.M. Isaacs b , P. Spiga c a
Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA b University of Wisconsin, Department of Mathematics, 480 Lincoln drive, Madison, WI 53706, USA c University of Milano-Bicocca, Dipartimento di Matematica Pura e Applicata, Via Cozzi 55, 20125 Milano, Italy
a r t i c l e
i n f o
Article history: Received 17 June 2015 Available online xxxx Keywords: Derangements Rank Permutation groups
a b s t r a c t Let G be a finite transitive group of rank r. We give a short proof that the proportion of derangements in G is at most 1 − 1/r and we classify the permutation groups attaining this bound. © 2015 Elsevier Inc. All rights reserved.
Let G be a finite transitive group on the set Ω. An element g ∈ G is called a derangement if g has no fixed points on Ω, that is, αg = α for each α ∈ Ω. Moreover, the rank of G is the number of orbits of G in its natural component-wise action on the Cartesian product Ω × Ω. The group action is said to be regular if the identity is the only element of G fixing an element of Ω.
E-mail addresses: [email protected] (R. Guralnick), [email protected] (I.M. Isaacs), [email protected] (P. Spiga). 1 Guralnick was partially supported by NSF grant DMS-1302886. http://dx.doi.org/10.1016/j.jcta.2015.07.003 0097-3165/© 2015 Elsevier Inc. All rights reserved.
R. Guralnick et al. / Journal of Combinatorial Theory, Series A 136 (2015) 198–200
199
The main result of this paper is the following. Theorem 1. If G is a finite transitive group on Ω of rank r, then the proportion of elements of G which are derangements on Ω is at most 1 − 1/r. Moreover, this upper bound is met if and only if G acts regularly. In light of Theorem 1, one might wonder whether, by excluding the family of regular permutation groups, the upper bound 1 − 1/r can be improved further. This is not the case. For example, in the group PSL2 (q) in its natural 2-transitive action, the proportion 1 1 of derangements is 12 − q+1 when q is odd and 12 − 2(q+1) when q is even, and this number tends to 1/2 as q tends to infinity. In the Suzuki group 2 B2 (q) and in the Ree group 2 G2 (q) in their natural 2-transitive actions on the points of a Steiner system, the proportion of q 2 +1 1 derangements are 21 − 2(qq+1 2 +1) and 2 − q 3 +1 (respectively), and both of these ratios tend to 1/2 as q tends to infinity. Similar examples can be obtained for larger ranks. The proof of Theorem 1 uses some character theory and is very much in the same spirit as the proof of the beautiful Cameron–Cohen lower bound (r − 1)/|Ω| for the proportion of derangements [2]. This lower bound has motivated a conjecture due independently to Shalev and Boston et al. [1], and in turn this conjecture has spurred some interesting research culminating with a complete solution by the first author and Fulman [4,7,6,5]. The first half of Theorem 1 was also proved in [3, Theorem 3.1] using the so-called second moment method, which is a standard result in probability theory. (The proof of [3, Theorem 3.1] is also very short.) Our proof only uses the Cauchy–Schwarz inequality and characterises the groups attaining the bound. Proof of Theorem 1. Let π be the permutation character of G and let D be the set of derangements of G. Observe that π, π = r and π(g) = 0 for every g ∈ D, and hence
|G|r =
π(g)2 =
g∈G
g∈G\D
⎛ ⎞2 1 ⎝ π(g)2 ≥ π(g)⎠ , |G| − |D| g∈G\D
where the last inequality is a direct application of the Cauchy–Schwarz inequality. As π(g) = 0 for every g ∈ D, this yields ⎛ |G|r ≥
1 ⎝ |G| − |D|
g∈G
⎞2 π(g)⎠ =
|G|2 |G| − |D|
and the first half of the theorem follows. Suppose now that |D|/|G| = 1 − 1/r. By chasing the previous series of inequalities, 2 we get g∈G\D π(g)2 = g∈G\D π(g) . In turn, the Cauchy–Schwarz inequality gives π(g) = π(h) for every g, h ∈ G \ D, that is, π(g) = π(1) = |Ω| for every g ∈ G \ D. Since
200
R. Guralnick et al. / Journal of Combinatorial Theory, Series A 136 (2015) 198–200
G acts faithfully, we must have G \ D = {1} and hence the identity of G is the only permutation fixing a point, that is, G is regular. 2 References [1] N. Boston, W. Dabrowski, T. Foguel, et al., The proportion of fixed-point-free elements in a transitive group, Comm. Algebra 21 (1993) 3259–3275. [2] P.J. Cameron, A.M. Cohen, On the number of fixed-point-free elements in a permutation group, Discrete Math. 106 (1992) 135–138. [3] P. Diaconis, J. Fulman, R. Guralnick, On fixed points of permutations, J. Algebraic Combin. 28 (2008) 189–218. [4] J. Fulman, R. Guralnick, Derangements in simple and primitive groups, in: Groups, Combinatorics, and Geometry, Durham, 2001, World Sci. Publ., River Edge, NJ, 2003, pp. 99–121. [5] J. Fulman, R. Guralnick, Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements, Trans. Amer. Math. Soc. 364 (2012) 3023–3070. [6] J. Fulman, R. Guralnick, Derangements in subspace actions of finite classical groups, Trans. Amer. Math. Soc. (2015), in press. [7] J. Fulman, R. Guralnick, Derangements in finite classical groups for actions related to extension field and imprimitive subgroups and the solution of the Boston–Shalev conjecture, preprint.