The depth of a finite simple group

Burness, Timothy C.; Liebeck, Martin W.; Shalev, Aner

doi:10.1090/proc/13937

Cited by 8 publications

(58 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The depth of finite solvable groups was studied by Kohler [14], and more generally by Shareshian and Woodroofe [19] in relation to lattice theory. We refer the reader to [3,5] for recent work on the length and depth of finite groups and finite simple groups. In [4], we extended these notions to algebraic groups.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The length and depth of compact Lie groups

2019

Self Cite

View full text Add to dashboard Cite

Let G be a connected Lie group. An unrefinable chain of G is a chain of subgroups G = G0 > G1 > · · · > Gt = 1, where each Gi is a maximal connected subgroup of Gi−1. In this paper, we introduce the notion of the length (respectively, depth) of G, defined as the maximal (respectively, minimal) length of such a chain, and we establish several new results for compact groups. In particular, we compute the exact length and depth of every compact simple Lie group, and draw conclusions for arbitrary connected compact Lie groups G. We obtain best possible bounds on the length of G in terms of its dimension, and characterize the connected compact Lie groups that have equal length and depth. The latter result generalizes a well known theorem of Iwasawa for finite groups. More generally, we establish a best possible upper bound on dim G ′ in terms of the chain difference of G, which is its length minus its depth.

show abstract

Section: Introductionmentioning

confidence: 99%

“…Note that these parameters are independent of any choice of isogeny type. In particular, we may (and will) always assume that if G is a nonabelian compact connected Lie group, then G ′ = i S i is a commuting product of simple groups given in (3).…”

Section: Introductionmentioning

confidence: 99%

The length and depth of compact Lie groups

2019

Self Cite

View full text Add to dashboard Cite

show abstract

“…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(A n ) 23 for all n, whereas l(A n ) tends to infinity with n).…”

Section: Introductionmentioning

confidence: 99%

“…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(A n ) 23 for all n, whereas l(A n ) tends to infinity with n). Upper bounds on the depth of each simple group of Lie type over F q are presented in [4,Theorem 4]; the bounds are given in terms of k, where q = p k with p a prime.…”

Section: Introductionmentioning

confidence: 99%

The length and depth of algebraic groups

2018

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…The length of G, denoted ℓ(G), is the maximum length of an unrefinable chain, and the depth of G, denoted λ(G), is the minimum length of an unrefinable chain. By [4], a nonabelian simple group G satisfies λ(G) (1 + o(1)) ℓ(G) log 2 (ℓ(G)) .…”

Section: Introductionmentioning

confidence: 99%

Classifying uniformly generated groups

Glasby

2019

Communications in Algebra

View full text Add to dashboard Cite

A finite group G is called uniformly generated, if whenever there is a (strictly ascending) chain of subgroups 1 < x 1 < x 1 , x 2 < · · · < x 1 , x 2 , . . . , x d = G, then d is the minimal number of generators of G. Our main result classifies the uniformly generated groups without using the simple group classification. These groups are related to finite projective geometries by a result of Iwasawa on subgroup lattices.

show abstract

The depth of a finite simple group

Cited by 8 publications

References 30 publications

The length and depth of compact Lie groups

The length and depth of compact Lie groups

The length and depth of algebraic groups

Classifying uniformly generated groups

Contact Info

Product

Resources

About