Chains of Subgroups in Groups of Lie Type Ii

Seitz, Gary M.; Solomon, Ronald; Turull, Alexandre

doi:10.1112/jlms/s2-42.1.93

Cited by 11 publications

(7 citation statements)

References 6 publications

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“…The maximum length of a chain of subgroups of a finite group G is called the length of G. This invariant arises naturally in several different contexts and it has been the subject of numerous papers since the 1960s (see [6,9,11,12,17,20,21], for example). The dual notion of depth is defined to be the minimal length of a chain of subgroups…”

Section: Introductionmentioning

confidence: 99%

The length and depth of compact Lie groups

2019

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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.

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Section: Introductionmentioning

confidence: 99%

The length and depth of compact Lie groups

2019

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“…The result cited in the preceding paragraph deals with the alternating groups. The problem was further considered by Seitz, Solomon and Turull [30,23]. It is not completely solved for all finite simple groups, but we can say that it is reasonably well understood.…”

Section: Subgroup Chains In Groupsmentioning

confidence: 99%

Chains of subsemigroups

et al. 2017

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We investigate the maximum length of a chain of subsemigroups in various classes of semigroups, such as the full transformation semigroups, the general linear semigroups, and the semigroups of order-preserving transformations of finite chains. In some cases, we give lower bounds for the total number of subsemigroups of these semigroups. We give general results for finite completely regular and finite inverse semigroups. Wherever possible, we state our results in the greatest generality; in particular, we include infinite semigroups where the result is true for these.The length of a subgroup chain in a group is bounded by the logarithm of the group order. This fails for semigroups, but it is perhaps surprising that there is a lower bound for the length of a subsemigroup chain in the full transformation semigroup which is a constant multiple of the semigroup order.

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“…For example, Babai [3] investigated the length of symmetric groups in relation to the computational complexity of algorithms for finite permutation groups. In a different direction, Seitz, Solomon and Turull studied the length of finite groups of Lie type in a series of papers in the early 1990s [31,33,34], motivated by applications to fixed-point-free automorphisms of finite soluble groups. In fact, the notion predates both the work of Babai and Seitz et al Indeed, Janko and Harada studied the simple groups of small length in the 1960s, culminating in Harada's description of the finite simple groups of length at most 7 in [17].…”

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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Chains of Subgroups in Groups of Lie Type Ii

Cited by 11 publications

References 6 publications

The length and depth of compact Lie groups

The length and depth of compact Lie groups

Chains of subsemigroups

On the length and depth of finite groups

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