Groupes et algébres de lie

Introduction to lie algebras and representation theory

1. (i) Suppose K is a conjugacy class of Sn contained in An; then K is called split if K is a union of two conjugacy classes of An. Show that the number of split conjugacy classes contained in An is equal to the number of characters χ ∈ Irr(Sn) such that χAn is not irreducible. (Hint. Consider the vector space of class functions on An which are invariant under conjugation by the transposition (12).)

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Character Theory of Finite Groups

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/63B506EB85D6875966CE63BA0336BC2F/S0025557200190515a.pdf/div-class-title-span-class-italic-lie-groups-and-lie-algebras-chapters-1-3-span-by-n-bourbaki-pp-450-dm-98-1989-isbn-3-540-50218-1-springer-div.pdf

Lie groups and Lie algebras (chapters 1-3 ), by N. Bourbaki. Pp 450. DM 98. 1989. ISBN 3-540-50218-1 (Springer)

In this window, all groups are assumed finite. Here we collect a number of results that play a significant role in the book (further material of an elementary nature that we sometimes take for granted is easily available in textbooks such as [H], [R] and [A]).

Finite Group Theory

Let G be a permutation group on a finite set !. A base for G is a subset B ! ! with pointwise stabilizer in G that is trivial; we write b(G) for the smallest size of a base for G. In this paper we prove that b(G) ! 6 if G is an almost simple group of exceptional Lie type and ! is a primitive faithful G-set. An important consequence of this result, when combined with other recent work, is that b(G) ! 7 for any almost simple group G in a non-standard action, proving a conjecture of Cameron. The proof is probabilistic and uses bounds on fixed point ratios.

/pdf/base-sizes-for-simple-groups-and-a-conjecture-of-cameron-4ep1bcmosq.pdf

Base sizes for simple groups and a conjecture of Cameron

Let G be a permutation group acting on a set . A subset of is a base for G if its pointwise stabilizer in G is trivial. We write b(G) for the minimal size of a base for G. We determine the precise value of b(G) for every primitive almost simple sporadic group G, with the exception of two cases involving the Baby Monster group. As a corollary, we deduce that b(G) 6 7, with equality if and only if G is the Mathieu group M24 in its natural action on 24 points. This settles a conjecture of Cameron.

/pdf/base-sizes-for-sporadic-simple-groups-1bl1mt0ofj.pdf

Base sizes for sporadic simple groups

A base of a permutation group G on a set is a subset B of such that the pointwise stabilizer of B in G is trivial. The base size of G, denoted by b(G), is the minimal cardinality of a base. Let G = Sn or An acting primitively on a set with point stabilizer H. In this note we prove that if H acts primitively on {1, . . . , n}, and does not contain An, then b(G) = 2 for all n 13. Combined with a theorem of James, this completes the classification of primitive actions of alternating and symmetric groups which admit a base of size two

https://eprints.soton.ac.uk/145455/1/baselms2.pdf

On base sizes for symmetric groups

A classical theorem of Jordan states that every finite transitive permutation group contains a derangement. This existence result has interesting and unexpected applications in many areas of mathematics, including graph theory, number theory and topology. Various generalisations have been studied in more recent years, with a particular focus on the existence of derangements with special properties. Written for academic researchers and postgraduate students working in related areas of algebra, this introduction to the finite classical groups features a comprehensive account of the conjugacy and geometry of elements of prime order. The development is tailored towards the study of derangements in finite primitive classical groups; the basic problem is to determine when such a group G contains a derangement of prime order r, for each prime divisor r of the degree of G. This involves a detailed analysis of the conjugacy classes and subgroup structure of the finite classical groups.

Classical Groups, Derangements and Primes

Let G be a finite almost simple classical group and let Ω be a faithful primitive non-standard G-set. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. Let b(G) be the minimal size of a base for G. A well-known conjecture of Cameron and Kantor asserts that there exists an absolute constant c such that b(G) ≤ slant c for all such groups G, and the existence of such an undetermined constant has been established by Liebeck and Shalev. In this paper we prove that either b(G) ≤ 4, or G = U 6 (2) · 2, G ω = U 4 (3) · 2 2 and b(G) = 5. The proof is probabilistic, using bounds on fixed point ratios.

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Timothy C. Burness

Papers

Base sizes for simple groups and a conjecture of Cameron

Base sizes for sporadic simple groups

On base sizes for symmetric groups

Classical Groups, Derangements and Primes

On base sizes for actions of finite classical groups