The length and depth of algebraic groups

Math. Z. https://doi.org/10.1007/s00209-018-2101-6 Mathematische Zeitschrift The length and depth of algebraic groups Timothy C. Burness1 · Martin W. Liebeck2 · Aner Shalev3 Received: 10 January 2018 / Accepted: 12 April 2018 © The Author(s) 2018 Abstract Let G be a connected algebraic group. An unrefinable chain of G is a chain of subgroups G = G 0 > G 1 > · · · > G t = 1, where each G i is a maximal connected subgroup of G i−1 . We introduce the notion of the length (respectively, depth) of G, defined as the maximal (respectively, minimal) length of such a chain. Working over an algebraically closed field, we calculate the length of a connected group G in terms of the dimension of its unipotent radical Ru (G) and the dimension of a Borel subgroup B of the reductive quotient G/Ru (G). In particular, a simple algebraic group of rank r has length dim B +r , which gives a natural extension of a theorem of Solomon and Turull on finite quasisimple groups of Lie type. We then deduce that the length of any connected algebraic group G exceeds 21 dim G. We also study the depth of simple algebraic groups. In characteristic zero, we show that the depth of such a group is at most 6 (this bound is sharp). In the positive characteristic setting, we calculate the exact depth of each exceptional algebraic group and we prove that the depth of a classical group (over a fixed algebraically closed field of positive characteristic) tends to infinity with the rank of the group. Finally we study the chain difference of an algebraic group, which is the difference between its length and its depth. In particular we prove that, for any connected algebraic group G with soluble radical R(G), the dimension of G/R(G) is bounded above in terms of the chain difference of G. The third author acknowledges the hospitality and support of Imperial College, London, while part of this work was carried out. He also acknowledges the support of ISF Grant 1117/13 and the Vinik chair of mathematics which he holds. B Timothy C. Burness Martin W. Liebeck Aner Shalev 1 School of Mathematics, University of Bristol, Bristol BS8 1TW, UK 2 Department of Mathematics, Imperial College, London SW7 2BZ, UK 3 Institute of Mathematics, Hebrew University, 91904 Jerusalem, Israel 123 T. C. Burness et al. Mathematics Subject Classification Primary 20E32, 20E15 ; Secondary 20G15, 20E28 1 Introduction The length of a finite group G, denoted by l(G), is the maximum length of a chain of subgroups of G. This interesting invariant was the subject of several papers by Janko and Harada [10,13,14] in the 1960s, culminating in Harada’s description of the finite simple groups of length at most 7 in [10]. In more recent years, the notion of length has arisen naturally in several different contexts. For example, Babai [1] considered the length of the symmetric group Sn in relation to the computational complexity of algorithms for finite permutation groups (a precise formula for l(Sn ) was later determined by Cameron, Solomon and Turull in [6]). Motivated by applications to fixed-point-free automorphisms of finite soluble groups, Seitz, Solomon and Turull studied the length of finite groups of Lie type in a series of papers in the early 1990s [21,23,24]. Let us highlight one of their main results, [24, Theorem A*], which states that if G = G r ( p k ) is a finite quasisimple group of Lie type and k is sufficiently large (with respect to the characteristic p), then l(G) = l(B) + r, (1) where B is a Borel subgroup of G and r is the twisted Lie rank of G. The dual notion of the depth of a finite group G, denoted by λ(G), is the minimal length of a chain of subgroups G = G 0 > G 1 > · · · > G t−1 > G t = 1, where each G i is a maximal subgroup of G i−1 . This invariant was studied by Kohler [15] for finite soluble groups and we refer the reader to more recent work of Shareshian and Woodroofe [22] for further results in the context of lattice theory. In [4] we proved several results on the depth of finite simple groups and we studied the relationship between the length and depth of simple groups (see [5] for further results on the length and depth of finite groups). For instance, [4, Theorem 1] classifies the simple groups of depth 3 (it is easy to see that λ(G)  3 for every non-abelian simple group G) and [4, Theorem 2] shows that alternating groups have bounded depth (more precisely, λ(An )  23 for all n, whereas l(An ) tends to infinity with n). Upper bounds on the depth of each simple group of Lie type over Fq are presented in [4, Theorem 4]; the bounds are given in terms of k, where q = p k with p a prime. In this paper, we extend the above notions of length and depth to connected algebraic groups over algebraically closed fields. Let G be a connected algebraic group over an algebraically closed field of characteristic p  0. An unrefinable chain of length t of G is a chain of subgroups G = G 0 > G 1 > · · · > G t−1 > G t = 1, where each G i is a maximal closed connected subgroup of G i−1 (that is, G i is maximal among the proper connected subgroups of G i−1 ). We define the length of G, denoted by l(G), to be the maximal length of an unrefinable chain. Similarly, the depth λ(G) of G is the minimal length of such a chain. Notice that we impose the condition that the subgroups in an unrefinable chain are connected, which seems to be the most natural (and interesting) definition in this setting. In the statements of our main results, and for the remainder of the paper, we assume that the given algebraic group is connected and the underlying field is algebraically closed (unless 123 The length and depth of algebraic groups stated otherwise). Also note that our results are independent of any choice of isogeny type. Our first result concerns the length of an algebraic group. Theorem 1 Let G be an algebraic group and let B be a Borel subgroup of the reductive group Ḡ = G/Ru (G). Then l(G) = dim Ru (G) + dim B + r, where r is the semisimple rank of Ḡ. Corollary 2 Let G be a simple algebraic group of rank r and let B be a Borel subgroup of G. Then l(G) = dim B + r. Moreover, every unrefinable chain of G of maximum length includes a maximal parabolic subgroup. The last sentence of the corollary is justified in Remark 3.1. By Lemma 2.2, the solubility of B implies that l(B) = dim B, so Corollary 2 is the algebraic group analogue of the aforementioned result of Solomon and Turull [24, Theorem A*] for finite quasisimple groups (see (1) above). Next, we relate the length of arbitrary algebraic groups G to their dimension. We clearly have l(G)  dim G. Theorem 3 Let G be an algebraic group. (i) l(G) > 21 dim G. (ii) l(G) = dim G if and only if G/R(G) ∼ = At1 for some t  0, where R(G) is the soluble radical of G. The lower bound in part (i) of Theorem 3 is essentially best possible. For example, if G = Cr is a symplectic group of rank r  1, then Corollary 2 implies that l(G) 1 3 1 = + → as r → ∞. dim G 2 4r + 2 2 We now tu (...truncated)