Variations of Landau’s theorem for p-regular and p-singular conjugacy classes

The well-known Landau’s theorem states that, for any positive integer k, there are finitely many isomorphism classes of finite groups with exactly k (conjugacy) classes. We study variations of this theorem for p-regular classes as well as p-singular classes. We prove several results showing that the structure of a finite group is strongly restricted by the number of p-regular classes or the number of p-singular classes of the group. In particular, if G is a finite group with O_p(G) = 1 then |G/F(G)|_p' is bounded in terms of the number of p-regular classes of G. However, it is not possible to prove that there are finitely many groups with no nontrivial normal p-subgroup and kp-regular classes without solving some extremely difficult number-theoretic problems (for instance, we would need to show that the number of Fermat primes is finite).     

On Reflection Subgroups of Finite Coxeter Groups

Let W be a finite Coxeter group. We classify the reflection subgroups of W up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup R of W the conjugacy class of its Coxeter elements to be injective, up to conjugacy.     

Groups whose vanishing class sizes are not divisible by a given prime

Let G be a finite group. An element ${g\in G}Let G be a finite group. An element g ? G{g\in G} is a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g) = 0: if this is the case, we say that the conjugacy class of g in G is a vanishing conjugacy class of G. In this paper we show that, if the size of every vanishing conjugacy class of G is not divisible by a given prime number p, then G has a normal p-complement and abelian Sylow p-subgroups.     

Remarks on the length of conjugacy classes of finite groups

Let G be a finite group. The question of how certain arithmetical conditions on the lengths of the conjugacy classes of G influence the group structure has been studied by several authors. In this paper we study restrictions on the structure of a finite group in which the lengths of conjugacy classes are not divisible by p ² for some prime p. We generalise and provide simplified proofs for some earlier results.     

A note on two Camina’s theorems on conjugacy class sizes

Let G be a finite group. We mainly investigate how certain arithmetical conditions on conjugacy class sizes of some elements of biprimary order of G influence the structure of G. Some known results are generalized.     

On the Diameter of a p-Regular Conjugacy Class Graph of Finite Groups II

Abstract

Let Gbe a finite p-solvable group. Let us consider the graph Γ* _p (G) whose vertices are the primes which occur as the divisors of the conjugacy classes of p-regular elements of G and two primes are joined by an edge if there exists such a class whose size is divisible by both primes. Suppose that Γ _p *(G) is a connected graph, then we prove that the diameter of this graph is at most 3 and this is the best bound.     

Sign conjugacy classes of the alternating groups

A conjugacy class C of a finite group G is a sign conjugacy class if every irreducible character of G takes value 0,1 or ?1 on C. In this paper, we classify the sign conjugacy classes of alternating groups.     

On the number of conjugacy classes of finite nilpotent groups

We establish the first super-logarithmic lower bound for the number of conjugacy classes of a finite nilpotent group. In particular, we obtain that for any constant c there are only finitely many finite p-groups of order pm with at most c⋅m conjugacy classes. This answers a question of L. Pyber.     

Products of primes in conjugacy class sizes and irreducible character degrees

We prove that if the product pq of distinct primes p and q divides the degree of some irreducible complex character of a finite group G, then pq divides the size of some conjugacy class of G.     

Regular CW-Complexes and Poset Resolutions of Monomial Ideals

Abstract

We study the generation of a finite group by its conjugacy classes, while generalizing basic concepts from linear algebra: basis and dimension. Besides the well known Burnside Basis Theorem for finite p-groups, there is no direct extension of these concepts to other families of finite groups. We show that by considering generating sets consisting of conjugacy classes, there is a possibility for such a generalization.     

On elements of order <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">s</Emphasis></Superscript> in the plane Cremona group over a field of characteristic <Emphasis Type="Italic">p</Emphasis>

We show that the plane Cremona group over a field of characteristic p > 0 does not contain elements of order of power of p larger than 2. We also describe conjugacy classes of elements of order p ². To the memory of Vasily Iskovskikh     

On the number of conjugacy classes of a finite solvable group II

We prove that a finite solvable group G has at least (49p+1)/60 conjugacy classes whenever p is a prime such that p² divides the order of G. We also construct an infinite family of finite solvable groups, where this bound is attained.     

Minimal and Maximal Length Involutions in Finite Coxeter Groups

Let W be a finite irreducible Coxeter group. The aim of this paper is to describe the maximal and minimal length elements in conjugacy classes of involutions in W.     

Minimal length elements in some double cosets of Coxeter groups

We study the minimal length elements in some double cosets of Coxeter groups and use them to study Lusztig's G-stable pieces and the generalization of G-stable pieces introduced by Lu and Yakimov. We also use them to study the minimal length elements in a conjugacy class of a finite Coxeter group and prove a conjecture in [M. Geck, S. Kim, G. Pfeiffer, Minimal length elements in twisted conjugacy classes of finite Coxeter groups, J. Algebra 229 (2) (2000) 570-600].     

Groups with Anticentral Elements

We study finite groups G with elements g such that |C _G (g)| = |G:G′|. (Such elements generalize fixed-point-free automorphisms of finite groups.) We show that these groups have a unique conjugacy class of nilpotent supplements for the commutator subgroup and, using the classification of finite simple groups, that these groups are solvable.     

On the number of conjugacy classes in a finite group II

In this work we obtain new properties connected with the number of conjugacy classes of elements of a finite group, through the analysis of the numberr _G(gN) of conjugacy classes of elements ofG that intersect the cosetgN, whereN is a normal subgroup ofG andg any element ofG. The results obtained about this number are not only used in the general problem of classifying finite groups according to the number of conjugacy classes, but they also allow us to improve and generalize known results relating to conjugacy classes due to P. Hall, M. Cartwright, A. Mann, G. Sherman, A. Vera-López and L. Ortíz de Elguea. Examples are given which illustrate our improvements. This work has been supported by the University of the Basque Country.     

The conjugacy vector of ap-group of maximal class

We find the conjugacy vector, i.e., we determine the number of conjugacy classes which compose the sets of the elements with centralizers of equal order, for several general families ofp-groups of maximal class which include those of order up top ⁹. As a consequence, we obtain the number of conjugacy classes,r(G), for the groups in these families. Also, we provide upper and lower bounds forr(G) and characterize when they are attained. Examples are given showing that the bounds are actually attained. This work has been supported by DGICYT grant PB91-0446 and by the University of the Basque Country.     

Finite Groups with a Disconnected p -Regular Conjugacy Class Graph

Abstract

Let G be a finite p-solvable group for a fixed prime p. Attach to G a graph Γ _p (G) whose vertices are the non-central p-regular conjugacy classes of G and connect two vertices by an edge if their cardinalities have a common prime divisor. In this note we study the structure and arithmetical properties of the p-regular class sizes in p-solvable groups G having Γ _p (G) disconnected.     

Characterization of the alternating groups by their order and one conjugacy class length

Let G be a finite group, and let N(G) be the set of conjugacy class sizes of G. By Thompson’s conjecture, if L is a finite non-abelian simple group, G is a finite group with a trivial center, and N(G) = N(L), then L and G are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation). In this article, we investigate validity of Thompson’s conjecture under a weak condition for the alternating groups of degrees p+1 and p+2, where p is a prime number. This work implies that Thompson’s conjecture holds for the alternating groups of degree p + 1 and p + 2.