On the length of finite simple groups having chain difference one

Petrillo, Joseph

doi:10.1007/s00013-006-1983-4

Cited by 3 publications

(4 citation statements)

References 5 publications

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“…This invariant was studied for finite simple groups. See [3,11,19] for the study of finite simple groups of chain difference one, and Corollary 9 in [4], where we bound the length of a finite simple group in terms of its chain difference. For algebraic groups we prove a stronger result, without assuming simplicity.…”

Section: Introductionmentioning

confidence: 99%

The length and depth of algebraic groups

2018

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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 R u (G) and the dimension of a Borel subgroup B of the reductive quotient G/R u (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 1 2 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.

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Section: Introductionmentioning

confidence: 99%

The length and depth of algebraic groups

2018

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“…The simple groups

G

with

cd (G) = 1

have been determined by Brewster et al . (also see and for related results).…”

Section: Introductionmentioning

confidence: 72%

“…There are also several papers on the so-called chain difference cd(G) = l(G) − λ(G) of a finite group G. For example, a well known theorem of Iwasawa [19] states that cd(G) = 0 if and only if G is supersoluble. The simple groups G with cd(G) = 1 have been determined by Brewster et al [7] (also see [18] and [29] for related results).…”

Section: Introductionmentioning

confidence: 99%

On the length and depth of finite groups

Burness

Liebeck

Shalev

2019

Proceedings of London Math Soc

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An unrefinable chain of a finite group G is a chain of subgroups G = G0 > G1 > · · · > Gt = 1, where each Gi is a maximal subgroup of Gi−1. The length (respectively, depth) of G is the maximal (respectively, minimal) length of such a chain. We studied the depth of finite simple groups in a previous paper, which included a classification of the simple groups of depth 3. Here, we go much further by determining the finite groups of depth 3 and 4. We also obtain several new results on the lengths of finite groups. For example, we classify the simple groups of length at most 9, which extends earlier work of Janko and Harada from the 1960s, and we use this to describe the structure of arbitrary finite groups of small length. We also present a numbertheoretic result of Heath-Brown, which implies that there are infinitely many non-abelian simple groups of length at most 9.Finally, we study the chain difference of G (namely the length minus the depth). We obtain results on groups with chain differences 1 and 2, including a complete classification of the simple groups with chain difference 2, extending earlier work of Brewster et al. We also derive a best possible lower bound on the chain ratio (the length divided by the depth) of simple groups, which yields an explicit linear bound on the length of

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“…The finite simple groups of minimal length 4 have depth 3 and chain difference 1, and so can be read off from Theorem 1 above, together with [4]. The precise list is given in [25,Theorem 3.2]. On the other hand, our results imply that the chain difference of a finite simple group is usually large.…”

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confidence: 85%

The depth of a finite simple group

Burness

Liebeck

Shalev

2018

Proc. Amer. Math. Soc.

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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.

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On the length of finite simple groups having chain difference one

Cited by 3 publications

References 5 publications

The length and depth of algebraic groups

The length and depth of algebraic groups

On the length and depth of finite groups

The depth of a finite simple group

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