Products Of Conjugacy Classes Research Articles

Products of conjugacy classes in SL2(k) and PSL2(k)

Let k be a field with u-invariant ≤ 2 . Assume further that k is not quadratically closed, char ( k ) ≠ 2 and | k | ≥ 5 . It is known that the covering number of both SL 2 ( k ) and PSL 2 ( k ) is three, while their extended covering number is four. In this article, we completely describe the product of two conjugacy classes in SL 2 ( k ) and PSL 2 ( k ) . Further, we also describe the product of three conjugacy classes (at least two of which are distinct) in SL 2 ( k ) and PSL 2 ( k ) .

The chiral ring of a symmetric orbifold and its large N limit

We analyze the chiral operator ring of the symmetric orbifold conformal field theory on the complex two-plane ℂ2. We compute the large N limit of the ring and exhibit its factorized leading order behaviour. We moreover calculate all structure constants at the subleading and sub-subleading order. These features are coded as properties of the symmetric group and we review the relevant mathematical theorems on the product of conjugacy classes in the center of the group algebra. We illustrate the efficiency of the formalism by iteratively computing broad classes of higher point extremal correlators. We point out generalizations of our simplest of models and argue that our combinatorial analysis is relevant to the organization of the large N perturbation theory of generic symmetric orbifolds.

Products of conjugacy classes in simple algebraic groups in terms of diagrams

For a simple algebraic group G over an algebraically closed field we study products of normal subsets. For this we mark the nodes of the Dynkin diagram of G. We use two types of labels, a binary marking and a labeling with nonnegative integers. The first is used to recognize large conjugacy classes which appear in a product of two conjugacy classes while the second is used to keep track of multiplicities of regular diagrams. In particular, we formulate sufficient conditions in terms of marked diagrams, for a product of normal subsets in G to contain regular semisimple elements.

Product of conjugacy classes of complete cycles in the alternating group

Product of conjugacy classes of complete cycles in the alternating group

Some problems about products of conjugacy classes in finite groups

‎We summarize several results about non-simplicity‎, ‎solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes‎. ‎We also collect some problems that have only been partially solved‎.

A note on the product of conjugacy classes of a finite group

In Guralnick and Moreto (Conjugacy classes, characters and products of elements, arXiv:1807.03550v1 , Theorem 4.2) it has been shown that if $$p\ne q$$ are two odd primes, $$\pi =\{2,p,q\}$$ and G is a finite group such that for every $$\pi $$ -elements $$x,y \in G$$ with $$(O(x),O(y))=1$$ , $$(xy)^G=x^Gy^G$$ , then G does not have any composition factors of order divisible by pq. In this note, inspired by the above result, we show that if p and q are two primes (not necessarily odd) and G is a finite group such that for every p-element x and q-element $$y \in G$$ , $$(xy)^G=x^Gy^G$$ , then G does not have any composition factors of order divisible by pq. In particular, we show that if p is an odd prime and G is a finite group such that for every p-element x and 2-element $$y \in G$$ , $$(xy)^G=x^Gy^G$$ , then G is p-solvable.

On Products of Irreducible Characters and Products of Conjugacy Classes in Finite Groups

Let G a finite group. For conjugacy classes A and B of G, the subset AB of G is a union of conjugacy classes of G, and for irreducible complex characters χ and ϕ of G, the product χϕ is a linear combination of irreducible characters of G with positive integer coefficients. In this article we give the structure of finite groups G whenever C 2 contains a central conjugacy class or does not contain a central conjugacy class of G, for all conjugacy classes C of G. Also in the case of (C −1) m C n ⊆ Z(G) the structure of G is given. Similar results when C is replaced by an irreducible character χ of G are discussed.

Products of conjugacy classes in finite and algebraic simple groups

We prove the Arad–Herzog conjecture for various families of finite simple groups — if A and B are nontrivial conjugacy classes, then AB is not a conjugacy class. We also prove that if G is a finite simple group of Lie type and A and B are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB is not a conjugacy class. We also prove a strong version of the Arad–Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product. A special case of this has been used by Prasad to prove a uniqueness result for Tits systems in quasi-reductive groups. Our final result is a generalization of the Baer–Suzuki theorem for p-elements with p≥5.

Products of conjugacy classes in simple groups

Let G be a finite group. For a ∈ G, let a G = {a g | g ∈ G} be the conjugacy class of a in G. In this paper, we study a conjecture due to Arad and Herzog which asserts that in a finite non-abelian simple group the product of two nontrivial conjugacy classes is never a single conjugacy class. In particular, we will verify this conjecture for several families of finite simple groups of Lie type.

Products of conjugacy classes and fixed point spaces

We prove several results on products of conjugacy classes in finite simple groups. The first result is that for any finite nonabelian simple groups, there exists a triple of conjugate elements with product11which generate the group. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann about the existence of elements in an irreducible linear group with small fixed space. We also show that there always exist two conjugacy classes in a finite nonabelian simple group whose product contains every nontrivial element of the group. We use this to show that every element in a nonabelian finite simple group can be written as a product of tworrth powers for any prime powerrr(in particular, a product of two squares answering a conjecture of Larsen, Shalev and Tiep).

Sums of adjoint orbits and [formula omitted]-singular dichotomy for [formula omitted

Let G be a real compact connected simple Lie group, and g its Lie algebra. We study the problem of determining, from root data, when a sum of adjoint orbits in g , or a product of conjugacy classes in G, contains an open set. Our general methods allow us to determine exactly which sums of adjoint orbits in su ( m ) and products of conjugacy classes in SU ( m ) contain an open set, in terms of the highest multiplicities of eigenvalues. For su ( m ) and SU ( m ) we show L 2 -singular dichotomy: The convolution of invariant measures on adjoint orbits, or conjugacy classes, is either singular to Haar measure or in L 2 .

On the product of conjugacy classes in unitary group and singular connections

We are attempting to give a new proof to the problem of characterization of the support of the product of conjugacy classes in the compact Lie group SU(n) without any reference to the Mehta-Seshadri theorem in algebraic geometry as it was the case in [1].

On the product of conjugacy classes in unitary group and singular connections

On the product of conjugacy classes in unitary group and singular connections

Derived length and products of conjugacy classes

Let G be a supersolvable group and A be a conjugacy class of G. Observe that for some integer η(AA −1) > 0, AA −1 = {ab −1: a, b ∈ A} is the union of η(AA −1) distinct conjugacy classes of G. Set C G (A) = {g ∈ G: a g = a for all a ∈ A. Then the derived length of G/C G (A) is less or equal than 2η(AA −1) − 1.

Products of conjugacy classes in perfect linear groups. Extended covering numbers

In this paper we obtain estimates of extended covering numbers for some classes of perfect linear groups. Bibliography: 19 titles.

Products of conjugacy classes in Chevalley groups over local rings

Products of conjugacy classes in Chevalley groups over local rings

Products of Conjugacy Classes in SU(2)

AbstractWe obtain a complete description of the conjugacy classes C1, … ,Cn in SU(2) with the property that C1 … Cn = SU(2). The basic instrument is a characterization of the conjugacy classes C1, … ,Cn+1 in SU(2) with C1 … Cn+1 ∋ I, which generalizes a result of [Je-We].

Content evaluation and class symmetric functions

In this article we study the evaluation of symmetric functions on the alphabet of contents of a partition. Applying this notion of content evaluation to the computation of central characters of the symmetric group, we are led to the definition of a new basis of the algebra Λ of symmetric functions over Q(n) that we call the basis of class symmetric functions. By definition this basis provides an algebra isomorphism between Λ and the Farahat–Higman algebra FH governing for all n the products of conjugacy classes in the center Z n of the group algebra of the symmetric group S n . We thus obtain a calculus of all connexion coefficients of Z n inside Λ. As expected, taking the homogeneous components of maximal degree in class symmetric functions, we recover the symmetric functions introduced by Macdonald to describe top connexion coefficients. We also discuss the relation of class symmetric functions to the asymptotic of central characters and of the enumeration of standard skew young tableaux. Finally we sketch the extension of these results to Hecke algebras.

Products of Conjugacy Classes in the Special Linear Groups#

This paper is concerned with products of conjugacy classes in the special linear groups SL n (F). We obtain a sufficient condition for the product of classes to cover SL n (F), depending on the shape of the rational normal form of matrices in the classes. One consequence of our result is that if the classes 𝒞 i in SL n (q) with 1 ≤ i ≤ khave the property that |𝒞 i | ≥ |SL n (q)|12then 𝒞 i = SL n (q). Our proof also gives a sharper bound on the extended covering number: any kgenerating conjugacy classes 𝒞 i in SL n (F) give 𝒞 i = SL n (F), provided that k ≥ 3n + 4.

Products of conjugacy classes in Chevalley groups I. Extended covering numbers

This paper is concerned with products of conjugacy classes in Chevalley groups. We prove that in any quasisimple Chevalley groupG proper or twisted, over any field, the extended covering number is bounded above linearly in terms of the rank ofG, that is, for some constante, for any Chevalley groupG, the product of anye · rank(G) non-central classes covers all ofG. We give estimates for the constante in different cases.