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Motivated in part by representation theoretic questions, we prove that if G is a finite quasi-simple group, then there exists an elementary abelian subgroup of G that contains a member of each conjugacy class of involutions of G.

Let $G$ be a finite group of order divisible by a prime $p$. The number of $p$-regular and $p'$-regular conjugacy classes of $G$ is at least $2\sqrt{p-1}$. Also, the number of $p$-rational and $p'$-rational irreducible characters of $G$ is at least $2\sqrt{p-1}$. Along the way we prove a uniform lower bound for the number of $p$-regular classes in a finite simple group of Lie type in terms of its rank and size of the underlying field., 46 pages

If C is a class of groups, a C -injector of a finite group G is a subgroup V of G with the property that V ∩ K is a C -maximal subgroup of K for all subnormal subgroups K of G. The classical result of B. Fischer, W. Gaschutz and B. Hartley states the existence and conjugacy of F -injectors in finite soluble groups for Fitting classes F . We shall show that for groups of nilpotent length at most 4, F -injectors permute with the members of a Sylow basis in the group. We shall exhibit the construction of a Fitting class and a group of nilpotent length 5, which fail to satisfy the result and show that the bound is the best possible.

Motivated by a 2004 conjecture by the author and J. Sangroniz, Y. Yang has recently proved that if G is solvable then the index in G of the 8th term of the ascending Fitting series is bounded in terms of the largest number of zeros in a row in the character table of G. In this note, we prove this result for arbitrary finite groups and propose a stronger form of the 2004 conjecture. We conclude the paper showing some possible ways to prove this strengthened conjecture.

We prove the p-version of a conjecture raised by Isaacs et al. in Proc. AMS [8] on the degree of a primitive character that divides the size of some conjugacy class of a finite solvable group, thus generalizing Marchi's results [10] in J. Algebra 2020.

The affine general linear group acting on a vector space over a prime field is a well-understood mathematical object. Its elementary abelian regular subgroups have recently drawn attention in applied mathematics thanks to their use in cryptography as a way to hide or detect weaknesses inside block ciphers. This paper is focused on building a convenient representation of their elements which suits better the purposes of the cryptanalyst. Several combinatorial counting formulas and a classification of their conjugacy classes are given as well., Comment: to appear in Journal of Algebra

We continue the project started in [8] to describe the structure of the finite groups G of characteristic p in terms of their Baumann components and the conjugacy class Bau p ( G ) . The reduction theorem proved in [8] allows to assume that G has a unique Baumann component. In this paper we use this property to determine the isomorphism type of G / O p ( G ) and the action of G on Ω 1 ( Z ( O p ( G ) ) ) . In addition, we prove reduction theorems which allow to focus on groups G which satisfy G / O p ( G ) ≅ SL n ( q ) , Sp 2 n ( q ) or G 2 ( q ) and O p ( G ) ⩽ B for B ∈ Bau p ( G ) .

Let G be a finite group, p a prime, and IBr p ( G ) the set of irreducible p-Brauer characters of G. Let e ¯ p ( G ) be the largest integer such that p e ¯ p ( G ) divides χ ( 1 ) for some χ ∈ IBr p ( G ) . We show that | G : O p ( G ) | p ≤ p k e ¯ p ( G ) for an explicitly given constant k. We also study the analogous problem for the p-parts of the conjugacy class sizes of p-regular elements of finite groups.

This article is a contribution to the following problem: does there exist a Polish non-archimedean group (equivalently: automorphism group of a Fraisse limit) that is extremely amenable, and has ample generics. As Fraisse limits whose automorphism groups are extremely amenable must be ordered, i.e., equipped with a linear ordering, we focus on ordered Fraisse limits. We prove that automorphism groups of the universal ordered boron tree, and the universal ordered poset have a comeager conjugacy class but no comeager 2-dimensional diagonal conjugacy class. We formulate general conditions implying that there is no comeager conjugacy class, comeager 2-dimensional diagonal conjugacy class or non-meager 2-dimensional topological similarity class in the automorphism group of an ordered Fraisse limit. We also provide a number of applications of these results.

Brauer and Fowler noted restrictions on the structure of a finite group G in terms of the order of the centralizer of an involution t in G. We consider variants of these themes. We first note that for an arbitrary finite group G of even order, we have |G| is less than the number of conjugacy classes of the Fitting subgroup times the order of the centralizer to the fourth power of any involution in G. This result does require the classification of the finite simple groups. The groups SL(2,q) with q even shows that the exponent 4 cannot be replaced by any exponent less than 3. We do not know at present whether the exponent 4 can be improved in general, though we note that the exponent 3 suffices for almost simple groups G. We are however able to prove that every finite group $G$ of even order contains an involution u such that [G:F(G)] is less than the cube of the order of the centralizer of u. There is a dichotomy in the proof of this fact, as it reduces to proving two residual cases: one in which G is almost simple (where the classification of the finite simple groups is needed) and one when G has a Sylow 2-subgroup of order 2. For the latter result, the classification of finite simple groups is not needed (though the Feit-Thompson odd order theorem is). We also prove a very general result on fixed point spaces of involutions in finite irreducible linear groups which does not make use of the classification of the finite simple groups, and some other results on the existence of non-central elements (not necessarily involutions) with large centralizers in general finite groups. We also show (without the classification of finite simple groups) that if t is an involution in G and p is a prime divisor of [G:F(G)], then p is at most 1 plus the order of the centralizer of t (and this is best possible)., Comment: Version 2 consists of some reorganization and some new results

Let G be a finite group, and let cs ( G ) denote the set of sizes of the conjugacy classes of G. The prime graph built on cs ( G ) , that we denote by Δ ( G ) , is the (simple undirected) graph whose vertices are the prime divisors of the numbers in cs ( G ) , and two distinct vertices p, q are adjacent if and only if pq divides some number in cs ( G ) . A rephrasing of the main theorem in [8] is that the complement Δ ‾ ( G ) of the graph Δ ( G ) does not contain any cycle of length 3. In this paper we generalize this result, showing that Δ ‾ ( G ) does not contain any cycle of odd length, i.e., it is a bipartite graph. In other words, the vertex set V ( G ) of Δ ( G ) is covered by two subsets, each inducing a complete subgraph (a clique). As an immediate consequence, setting ω ( G ) to be the maximum size of a clique in Δ ( G ) , the inequality | V ( G ) | ≤ 2 ω ( G ) holds for every finite group G.

In this paper we introduce a new conjugacy class Bau p ( G ) of p-subgroups of finite groups G of characteristic p. We then prove some factorization and decomposition theorems related to this conjugacy class. In particular, these results show that the only obstructions for B ∈ Bau p ( G ) being normal in G are the Baumann components of G, a class of subnormal subgroups E with E / O p ( E ) quasisimple or SL 2 ( p ) ′ for p ≤ 3 .

All finite simple groups are determined with the property that every Galois orbit on conjugacy classes has size at most 4. From this we list all finite simple groups $G$ for which the normalized group of central units of the integral group ring ZG is an infinite cyclic group., Comment: 15 pages. Expanded version

We give a combinatorial formula to calculate the Euler characteristic of an analogue of a Deligne–Lusztig variety, denoted Y w , g , which is attached to an element w in the Weyl group of G L n and g ∈ G L n . The main theorem of this paper states that the Euler characteristic of Y w , g only depends on the unipotent part of the Jordan decomposition of g and the conjugacy class of w. It generalizes the formula of the Euler characteristic of Springer fibers for type A.

We study intersections of the form g 1 C 1 ∩ g 2 C 2 , where C i are conjugacy classes of arbitrary finite simple groups and g i are group elements. We show that, generically, | g 1 C 1 ∩ g 2 C 2 | ∼ | C 1 | | C 2 | / | G | , which means that the events g 1 C 1 , g 2 C 2 are almost independent in G. We also discuss the dimension and the irreducibility of such intersections in simple algebraic groups, and expose the anomaly of SL 2 . This work is motivated by recent questions of Hrushovski.

Let G be a linear algebraic group over an algebraically closed field of characteristic p ≥ 0 . We show that if H 1 and H 2 are connected subgroups of G such that H 1 and H 2 have a common maximal unipotent subgroup and H 1 / R u ( H 1 ) and H 2 / R u ( H 2 ) are semisimple, then H 1 and H 2 are G-conjugate. Moreover, we show that if H is a semisimple linear algebraic group with maximal unipotent subgroup U then for any algebraic group homomorphism σ : U → G , there are only finitely many G-conjugacy classes of algebraic group homomorphisms ρ : H → G such that ρ | U is G-conjugate to σ. This answers an analogue for connected algebraic groups of a question of B. Kulshammer. In Kulshammer's original question, H is replaced by a finite group and U by a Sylow p-subgroup of H; the answer is then known to be no in general. We obtain some results in the general case when H is non-connected and has positive dimension. Along the way, we prove existence and conjugacy results for maximal unipotent subgroups of non-connected linear algebraic groups. When G is reductive, we formulate Kulshammer's question and related conjugacy problems in terms of the nonabelian 1-cohomology of unipotent radicals of parabolic subgroups of G, and we give some applications of this cohomological approach. In particular, we analyse the case when G is a semisimple group of rank 2.

[EN] We obtain some structural properties of a factorised group G = AB, given that the conjugacy class sizes of certain elements in A boolean OR B are not divisible by p(2), for some prime p. The case when G = AB is a mutually permutable product is especially considered., The first author is supported by Proyecto Prometeo II/2015/011, Generalitat Valenciana (Spain), and the second author is supported by Proyecto MTM2014-54707-C3-1-P, Ministerio de Economia, Industria y Competitividad (Spain). The results in this paper are part of the third author's Ph.D. thesis, and he acknowledges the predoctoral grant ACIF/2016/170, Generalitat Valenciana (Spain).

Let G be a finite group. An element g of G is called a vanishing element if there exists an irreducible character χ of G such that χ ( g ) = 0 ; in this case, we say that the conjugacy class of g is a vanishing conjugacy class. In this paper, we discuss some arithmetical properties concerning the sizes of the vanishing conjugacy classes in a finite group.

Given a prime number $p$, every irreducible character $\chi$ of a finite group $G$ determines a unique conjugacy class of $p$-subgroups of $G$ which we will call the anchors of $\chi$. This invariant has been considered by L. Barker in the context of finite $p$-solvable groups. Besides proving some basic properties of these anchors, we investigate the relation to other $p$-groups which can be attached to irreducible characters, such as defect groups, vertices in the sense of J. A. Green and vertices in the sense of G. Navarro.

This paper discusses two related questions. First, given a G-invariant character θ of a normal subgroup N of a solvable group, what can we say if the number of characters of G above θ is in some sense as small as possible? Isaacs and Navarro [5] have shown that under certain assumptions about primes dividing the order of the group, one can show that G / N must have a very particular structure. Here we show that these assumptions can be weakened to obtain results about all solvable groups. We also discuss a related question about blocks. For a prime p and a p-block B of G, we let k ( B ) denote the number of ordinary characters in B. It is relatively easy to show that k ( B ) is bounded below by k ( G , D ) , which is the number of conjugacy classes of G that intersect the defect group D of B. In this paper we ask what can be said if equality is achieved. We show that for p-solvable groups, if k ( B ) = k ( G , D ) , then B is nilpotent and thus k ( B ) = | Irr ( D ) | . In addition, we show that this result holds for many blocks of arbitrary finite groups, including all blocks of the symmetric groups. We also extend a result on fully ramified coprime actions in [5] .

Assume k is a positive integer and p is a prime. Let ν ( G ) be the number of conjugacy classes of nonnormal subgroups of a finite group G and N C N ( p , k ) = { ν ( G ) ∈ [ 0 , k p ] | G is a finite p -group } . In this paper, the set N C N ( p , k ) is determined for k ≤ 2 , and it is discovered that there is a new gap in the values that ν ( G ) can take in the case of finite p-groups. In particular, the number of conjugacy classes of nonnormal subgroups of minimal non-abelian p-groups is determined.

A group G is called subgroup conjugacy separable if for every pair of non-conjugate finitely generated subgroups of G, there exists a finite quotient of G where the images of these subgroups are not conjugate. We prove that limit groups are subgroup conjugacy separable. We also prove this property for one relator groups of the form R = 〈 a 1 , . . . , a n | W n 〉 with n > | W | . The property is also proved for infinite virtual retracts (equivalently for infinite quasiconvex subgroups) of hyperbolic virtually special groups.

Let k be an algebraically closed field and L = k ( ( ϵ ) ) the field of Laurent series over k. Let G be a connected reductive group over k such that the characteristic of k does not divide the order of the Weyl group of G. For q ∈ k × , the automorphism of L, defined by ∑ a i ϵ i ↦ ∑ a i ( q ϵ ) i , induces an automorphism σ q of the loop group G ( L ) . We show that, when q is not a root of unity, the classification of σ q -conjugacy classes of G ( L ) can be reduced to the classification of unipotent classes of (non-connected) reductive groups. As an application, we recover the classification (of σ q -conjugacy classes) for G = G L n .

We show that the Sylow p-subgroups of a symmetric group, respectively an alternating group, are characterized as the p-subgroups containing all elementary abelian p-subgroups up to conjugacy of the symmetric group, respectively the alternating group. We apply the characterization result for symmetric groups to compute the vertices of the hook Specht modules associated to the partition ( k p − p , 1 p ) under the assumption that k ≡ 1 mod p and k ≢ 1 mod p 2 .

For a prime $p$, we say that a conjugacy class of a finite group $G$ is $p$-vanishing if every irreducible character of $G$ of degree divisible by $p$ takes value 0 on that conjugacy class. In this paper we completely classify 2-vanishing and 3-vanishing conjugacy classes for the symmetric group and do some work in the classification of $p$-vanishing conjugacy classes of the symmetric group for $p\geq 5$. This answers a question by Navarro for $p=2$ and $p=3$ and partly answers it for $p\geq 5$.

The aim of this paper is to investigate the behaviour of uncountable groups of regular cardinality ℵ in which all proper subgroups of cardinality ℵ belong to a given group class X . It is proved that if every proper subgroup of G of cardinality ℵ has finite conjugacy classes, then also the conjugacy classes of G are finite, provided that G has no simple homomorphic images of cardinality ℵ. Moreover, it turns out that if G is a locally graded group of cardinality ℵ in which every proper subgroup of cardinality ℵ contains a nilpotent subgroup of finite index, then G is nilpotent-by-finite, again under the assumption that G has no simple homomorphic images of cardinality ℵ. A similar result holds also for uncountable locally graded groups whose large proper subgroups are abelian-by-finite.

A closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup. In this paper we classify all irreducible A 1 subgroups of exceptional algebraic groups G. Consequences are given concerning the representations of such subgroups on various G-modules: for example, the conjugacy classes of irreducible A 1 subgroups are determined by their composition factors on the adjoint module of G.

Let G = E 7 ( q ) be the finite exceptional group of Lie type (universal version). We classify, up to conjugacy, all maximal-proper 2-local subgroups of G , that is, all 2-local M G which are maximal with respect to inclusion among all proper subgroups of G which are 2-local. For this purpose, we first determine, up to conjugacy, all elementary-abelian 2-subgroups of G containing the center Z ( G ) . These classifications are an important first step towards a classification of the 2-radical subgroups of G .

Let S be a compact Riemann surface of genus g > 1 , and let τ : S → S be any anti-conformal automorphism of S, of order 2. Such an anti-conformal involution is known as a symmetry of S, and the species of all conjugacy classes of all symmetries of S constitute what is known as the symmetry type of S. The surface S is said to have maximal real symmetry if it admits a symmetry τ : S → S such that the compact Klein surface S / τ has maximal symmetry (which means that S / τ has the largest possible number of automorphisms with respect to its genus). If τ has fixed points, which is the only case we consider here, then the maximum number of automorphisms of S / τ is 12 ( g − 1 ) . In the first part of this paper, we develop a computational procedure to compute the symmetry type of every Riemann surface of genus g with maximal real symmetry, for given small values of g > 1 . We have used this to find all of them for 1 g ≤ 101 , and give details for 1 g ≤ 25 (in an appendix). In the second part, we determine the symmetry types of four infinite families of Riemann surfaces with maximal real symmetry. We also determine the full automorphism group of the Klein surface S / τ associated with each symmetry τ : S → S .

Let G be a connected algebraic group and consider the adjoint action μ : G × G → G defined by μ ( g , x ) = g x g − 1 . We show that this action is observable. A major step in our proof is to show that the adjoint action is induced generically from the conjugating action of N G ( T ) (the normalizer of a maximal torus) on a certain open subset of C G ( T ) (the centralizer of T). As a direct consequence we observe that the ring of invariants k [ G ] G ⊂ k [ G ] separates the conjugacy classes of G generically.

We show that Nichols algebras of most simple Yetter–Drinfeld modules over the projective special linear group over a finite field, corresponding to non-semisimple orbits, have infinite dimension. We spell out a new criterium to show that a rack collapses., Facultad de Ciencias Exactas

Let acd(G) and acs(G) denote the average irreducible character degree and the average conjugacy class size, respectively, of a finite group G. The main objective of this paper is to find the lower bounds of acd(G) and acs(G) for a non-r-solvable group G where r is a prime. We show that PSL(2,r), where r≥5, is a non-r-solvable group with the smallest average character degree and the smallest average class size.

We call an irreducible character p-singular if p divides its degree. We prove a number of equivalent conditions for a character of the symmetric group S n to be p-singular, involving a certain family of conjugacy classes. This answers in part a question by Navarro.

We improve earlier inequalities relating the nilpotence class of a p -group to the sizes of its conjugacy classes.

Let G be a connected reductive algebraic group defined over an algebraically closed field k. The aim of this paper is to present a method to find triples ( G , M , H ) with the following three properties. Property 1: G is simple and k has characteristic 2. Property 2: H and M are closed reductive subgroups of G such that H M G , and ( G , M ) is a reductive pair. Property 3: H is G-completely reducible, but not M-completely reducible. We exhibit our method by presenting a new example of such a triple in G = E 7 . Then we consider a rationality problem and a problem concerning conjugacy classes as important applications of our construction.

Let g be a simple Lie algebra of type F 4 or E n defined over a local or global field k of characteristic zero. We show that g can be obtained by the Tits construction from an octonion algebra O and a cubic Jordan algebra J . In particular, g contains a dual pair h defined over k which is the direct sum of the derivation algebras of O and J . We determine the conjugacy classes of k-forms of h in g .

We consider the finite exceptional groups of Lie type E 6 + 1 ( q ) = E 6 ( q ) and E 6 − 1 ( q ) = E 6 2 ( q ) , both the universal versions. We classify, up to conjugacy, the maximal p-local subgroups and radical p-subgroups of G = E 6 e ( q ) for p ⩾ 5 with p ∤ q and q ≡ e mod p , and for p = 3 with 3 ∤ q and q ≡ − e mod 3 . As an application, the essential p-rank of the Frobenius category F D ( G ) is determined, where D is a Sylow p-subgroup of G. Moreover, if p = 3 , then we show that there is a subgroup H = F 4 ( q ) of G containing D such that F D ( G ) = F D ( H ) , that is, H controls 3-fusion in G.

In this paper we give a bound to the number of conjugacy classes of maximal subgroups of any almost simple group whose socle is a classical group of Lie type. The bound is 2 n 5.2 + n log 2 log 2 q , where n is the dimension of the classical socle and q is the size of the defining field. To obtain the bound, we first bound the number of projective cross-characteristic representations of simple groups of Lie type as a function of the representation degree. These bounds are computed for different families of groups separately. In the computation, we use information on conjugacy class numbers, minimal character degrees and gaps between character degrees.

Let W be a Coxeter group. In this paper, we establish that, up to going to some finite index normal subgroup W_0 of W, any two cyclically reduced expressions of conjugate elements of W_0 only differ by a sequence of braid relations and cyclic shifts. This thus provides a simple description of conjugacy classes in W_0. As a byproduct of our methods, we also obtain a characterisation of straight elements of W, namely of those elements w in W for which $\ell(w^n)=n\ell(w)$ for any integer n. In particular, we generalise previous characterisations of straight elements within the class of so-called cyclically fully commutative (CFC) elements, and we give a shorter and more transparent proof that Coxeter elements are straight., 12 pages, to appear in Journal of Algebra

This paper deals with the problem of conjugacy of Cartan subalgebras for affine Kac-Moody Lie algebras. Unlike the methods used by Peterson and Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of J. Tits on buildings., 24 pages

The action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been several attempts to extend the notion of conjugacy to semigroups. In this paper, we present a new definition of conjugacy that can be applied to an arbitrary semigroup and it does not reduce to the universal relation in semigroups with a zero. We compare the new notion of conjugacy with existing definitions, characterize the conjugacy in various semigroups of transformations on a set, and count the number of conjugacy classes in these semigroups when the set is infinite., Comment: 41 pages, 14 figures

A group is boundedly simple if, for some constant N, every nontrivial conjugacy class generates the whole group in N steps. For a large class of trees, Tits proved simplicity of a canonical subgroup of the automorphism group, which is generated by pointwise stabilizers of edges. We prove that only for uniform subdivisions of biregular trees are such groups boundedly simple. In fact these groups are 8-boundedly simple. As a consequence, we prove that if G is boundedly simple and G acts by automorphisms on a tree, then G fixes some vertex of A, or stabilizes some end of A, or the smallest nonempty G-invariant subtree of A is a uniform subdivision of a biregular tree.