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A weak Cayley table isomorphism is a bijection φ : G → H of groups such that φ(xy) ~ φ(x)φ(y) for alt x,y e G. Here ~ denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ : G → G forms a group W(G) that contains the j automorphism group Aut(G) and the inverse map I : G → G,x t-> x-1. Let W0(G) = (Aut(G), I) < W(G) and say that G has triviaI weak Cayley table group if W(G) = W0(G). We show that all finite irreducible Coxeter groups (except possibly E8) have trivial weak Cayley table group, as well as most alternating groups. We also consider some sporadic simple groups. [ABSTRACT FROM AUTHOR]

A group is called a V-group if it has finite conjugacy classes of subnormal subgroups. It is proved here that if G is a periodic soluble group in which every subnormal subgroup of infinite rank has finitely many conjugates, then G is a V -group, provided that its Hirsch–Plotkin radical has infinite rank. Corresponding results for periodic soluble groups in which every subnormal subgroup of infinite rank has finite index in its normal closure and for those in which every subnormal subgroup of infinite rank is finite over its core, are also obtained. Moreover, it is shown that the assumption on the Hirsch–Plotkin radical can be avoided in the case of periodic groups with nilpotent commutator subgroup. [ABSTRACT FROM AUTHOR]

It is proved here that if G is a locally graded group satisfying the minimal condition on subgroups which are not locally supersoluble, then G is either locally supersoluble or a Cernikov group. The same conclusion holds for locally finite groups satisfying the weak minimal condition on non-(locally supersoluble) subgroups. As a consequence, it is shown that any in nite locally graded group whose non-(locally supersoluble) subgroups lie into finitely many conjugacy classes must be locally supersoluble [ABSTRACT FROM AUTHOR]

For a finite group G, let γ(G) denote the number of conjugacy classes of all non-nilpotent subgroups of G, and let π(G) denote the set of the prime divisors of |G|. In this paper, we establish lower bounds on γ(G). In fact, we show that if G is a finite solvable group, then γ(G) = 0 or γ(G) ≥ 2|π(G)|-2, and if G is non-solvable, then γ(G) ≥ |π(G)| + 1. Both lower bounds are best possible. [ABSTRACT FROM AUTHOR]

LetGbe a finite group and τ(G) denote the number of conjugacy classes of all non-abelian subgroups ofG. The symbol π(G) denotes the set of the prime divisors of |G|. In this paper, finite groups with τ(G) ≤ |π(G)| are classified completely. Furthermore, finite nonsolvable groups with τ(G) = |π(G)| +1 are determined. [ABSTRACT FROM AUTHOR]

Let Fpn G be the group algebra of a finite p-group G over a field FPn with pn elements, and V(FPnG) the subgroup of units of augmentation 1. We investigate the conjugacy classes of V(Pn G), showing that p2n divides the order of the conjugacy class ∣ Ca ∣ for a noncentral element a e Fpn G and V(Fpn G) contains a conjugacy class of order p 2n, if a nonabelian finite p-group G has a factor group G/H such that its centre is of index p2. [ABSTRACT FROM AUTHOR]

The work is inspired by an article of Herzog, Longobardi, and Maj, who considered groups with a finite number of infinite conjugacy classes. Their main results were obtained under assumption that theFC-center is of finite index in the group. We consider here infinite groups with a finite number of conjugacy classes of any size (FNCC-groups). Hence theFC-center in our case will be finite, but of infinite index in the group. Among results on these groups we give a criterion for a wreath product ofFNCC-groups to be anFNCC-group. [ABSTRACT FROM AUTHOR]

The theory of supercharacters, which generalizes classical character theory, was recently introduced by P. Diaconis and I.M. Isaacs, building upon earlier work of C. André. We study supercharacter theories on (Z/nZ)d induced by the actions of certain matrix groups, demonstrating that a variety of exponential sums of interest in number theory (e.g., Gauss, Ramanujan, Heilbronn, and Kloosterman sums) arise in this manner. We develop a generalization of the discrete Fourier transform, in which supercharacters play the role of the Fourier exponential basis. We provide a corresponding uncertainty principle and compute the associated constants in several cases. [ABSTRACT FROM AUTHOR]

Let G be a finite group and v(G) denote the number of conjugacy classes of non-normal subgroups of G. In this paper, all nilpotent groups G with v(G) = 3 are classified. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let x denote the conjugacy class of an element x of G. We classify all finite groups G in the following three cases: (i) Each non-trivial conjugacy class of G together with the identity element 1 is a subgroup of G, (ii) union of any two distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G, and (iii) union of any three distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G. [ABSTRACT FROM AUTHOR]

Finite nilpotent groups having exactly four conjugacy classes of non-normal subgroups are classified. [ABSTRACT FROM AUTHOR]

We show that every orbital measure, ${\mu }_{x} $, on a compact exceptional Lie group or algebra has the property that for every positive integer either ${ \mu }_{x}^{k} \in {L}^{2} $ and the support of ${ \mu }_{x}^{k} $ has non-empty interior, or ${ \mu }_{x}^{k} $ is singular to Haar measure and the support of ${ \mu }_{x}^{k} $ has Haar measure zero. We also determine the index $k$ where the change occurs; it depends on properties of the set of annihilating roots of $x$. This result was previously established for the classical Lie groups and algebras. To prove this dichotomy result we combinatorially characterize the subroot systems that are kernels of certain homomorphisms. [ABSTRACT FROM AUTHOR]

Let G be a finite group and ψ(G) = ∑g∈Go(g), where o(g) denotes the order of g ∈ G. In this short paper we show that the conjecture of minimality of ψ(G) in simple groups, posed in [ J. Algebra Appl. 10(2) (2011) 187-190], is incorrect. [ABSTRACT FROM AUTHOR]

LetGbe a finite group andM(G) be the subgroup ofGgenerated by all noncentral elements ofGthat lie in the conjugacy classes of the smallest size. Recently several results have been proved regarding the nilpotency class ofM(G) andF(M(G)), whereF(M(G)) denotes the Fitting subgroup ofM(G). We prove some conditional results regarding the nilpotency class ofM(G). [ABSTRACT FROM PUBLISHER]

Abstract: We introduce the notion of an ‘inverse property’ (IP) quandle which we propose as the right notion of ‘Lie algebra’ in the category of sets. For any IP-quandle we construct an associated group . For a class of IP-quandles which we call ‘locally skew’, and when is finite, we show that the noncommutative de Rham cohomology is trivial aside from a single generator θ that has no classical analogue. If we start with a group G then any subset which is ad-stable and inversion-stable naturally has the structure of an IP-quandle. If also generates G then we show that with central kernel, in analogy with the similar result for the simply-connected covering group of a Lie group. We prove that this ‘covering map’ is an isomorphism for all finite crystallographic reflection groups W with the set of reflections, and that is locally skew precisely in the simply laced case. This implies that when W is simply laced, proving in particular a conjecture for in Majid (2004) [12]. We also consider as a locally skew IP-quandle ‘Lie algebra’ of and show that , the braid group on 3 strands. The map which therefore arises naturally as a covering map in our theory, coincides with the restriction of the usual universal covering map to the inverse image of . [Copyright &y& Elsevier]

Let n be an even number and either q = 8 or q > 9. We prove a conjecture of Thompson (Problem 12.38 in [3]) for an infinite class of finite simple groups of Lie type. More precisely, if S ∈ { C( q), B( q)}, then every finite group G for which Z( G) = 1 and N( G) = N( S) will be isomorphic to S. Note that N( G) = { n : G has a conjugacy class of size n}. The main consequence of this result is showing the validity of AAM's conjecture (Problem 16.1 in [3]) for the groups under study. [ABSTRACT FROM AUTHOR]

In 2007, S. V. Tikhonov introduced a complete metric on the space of mixing transformations. This metric generates a topology called the leash topology. Tikhonov posed the following problem: what conditions should be satisfied by a mixing transformation T for its conjugacy class to be dense in the space of mixing transformations equipped with the leash topology. We show the conjugacy class to be dense for every mixing transformation T. As a corollary, we find that a generic mixing transformation is rank 1. [ABSTRACT FROM AUTHOR]

In this paper we introduce the conjugate graph associated to a nonabelian group G with vertex set G\Z(G) such that two distinct vertices join by an edge if they are conjugate. We show if , where S is a finite nonabelian simple group which satisfy Thompson's conjecture, then G ≅ S. Further, if central factors of two nonabelian groups H and G are isomorphic and |Z(G)| = |Z(H)|, then H and G have isomorphic conjugate graphs. [ABSTRACT FROM AUTHOR]

We consider the group G of R-automorphisms of the polynomial ring R[x] in the case where the ring R has nonzero nilpotent elements. Little is known about G in this case, and because of the importance of G in understanding questions involving the polynomial ring R[x], we initiate here several lines of investigation. We do this by examining in detail examples involving the ring of integers modulo n. If R is a local ring with maximal ideal m such that R/m = ℤ2 and m 2 = (0), we describe more explicitly the structure of G and determine all rings of invariants of R[x] with respect to subgroups of G. [ABSTRACT FROM PUBLISHER]

It is shown that if G is a group of even order with trivial center such that ΙGΙ > 2ΙCG(t)Ι3 for some involution t ∈ G, then there exists a proper subgroup H of G such that ΙGΙ < ΙHΙ2. If ΙGΙ > ΙCG(t)Ι3 and k(G) is the class number of G, then ΙGΙ ≤ k(G)3/Ι [ABSTRACT FROM AUTHOR]

Let 픛 be a class of groups. A group G is said to be minimal non-픛 if all proper subgroups of G are 픛-groups but G itself is not. The aim of this paper is to study the class of minimal non-FCn-groups, where FCn (n is a positive integer) is a class of generalized FC-groups introduced in [F. de Giovanni, A. Russo and G. Vincenzi, Groups with restricted conjugacy classes, Serdica Math. J.28 (2002) 241-254]. [ABSTRACT FROM AUTHOR]

As an important application of Thompson's theorem [9, Theorem 10.4.2], a finite group is solvable if it has an abelian maximal subgroup. In this article, we mainly investigate the influence of some quantitative properties of abelian subgroups on solvability of finite groups. Some new results are obtained. [ABSTRACT FROM PUBLISHER]

Abstract: It has been proved recently by Moretó (2007) and Craven (2008) that the order of a finite group is bounded in terms of the largest multiplicity of its irreducible character degrees. A conjugacy class version of this result was proved for solvable groups by Jaikin-Zapirain (2005) . In this note, we prove that if G is a finite simple group then the order of G, denoted by , is bounded in terms of the largest multiplicity of its conjugacy class sizes. As a consequence, we prove that if the largest multiplicity of conjugacy class sizes of any quotient of a finite group G is m, then is bounded in terms of m. [Copyright &y& Elsevier]

We prove that the automorphism group Aut(m,p,n) of an imprimitive complex reflection group G(m,p,n) is the product of a normal subgroup T(m,p,n) by a subgroup R(m,p,n), where R(m,p,n) is the group of automorphisms that preserve reflections and T(m,p,n) consists of automorphisms that map every element of G(m,p,n) to a scalar multiple of itself. [ABSTRACT FROM AUTHOR]

Abstract: Huber (1956)  considered the following problem on the hyperbolic plane . Consider a strictly hyperbolic subgroup of automorphisms on with compact quotient, and choose a conjugacy class in this group. Count the number of vertices inside an increasing ball, which are images of a fixed point under automorphisms in the chosen conjugacy class, and describe the asymptotic behaviour of this number as the size of the ball goes to infinity. We use a well-known analogy between the hyperbolic plane and the regular tree to solve this problem on the regular tree. [Copyright &y& Elsevier]

Abstract: The numbers of connected components of the spectrums of three kinds of rings of characters of a finite group are determined in terms of the numbers of special kinds of conjugacy classes of the group. [ABSTRACT FROM AUTHOR]

Let K be a conjugacy class of a finite p-group G where p is a prime, and let K-1 denote the conjugacy class of G consisting of the inverses of the elements in K. We observe that, in several cases, the number of elements in the product KK-1 is congruent to 1 modulo p - 1, and we pose the question in which generality this congruence is valid. We also consider related properties of the class multiplication constants of G. Furthermore, let χ be an irreducible character of G, and let [image omitted] denote the complex conjugate of χ. We show that, in several cases, the number of irreducible constituents of the product [image omitted] is congruent to 1 modulo p - 1, and we pose the question in which generality this congruence is valid. [ABSTRACT FROM AUTHOR]

In this article, we describe the summit sets in the 3-braid group, the smallest element in a summit set, and we compute the Hilbert series corresponding to conjugacy classes. The results are related to the Birman-Menasco classification of knots with braid index three or less than three. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study conjugacy classes of the symplectic Renner monoid , and find the following results. Each element in is a join of strictly disjoint cycles and strings. There is an explicit one-to-one correspondence between conjugacy classes and symplectic partitions. A formula for calculating the number of conjugacy classes is obtained. An explicit formula for computing the number of elements in each conjugacy class is given. A relationship between conjugacy classes and Munn conjugacy classes is described. [ABSTRACT FROM AUTHOR]

The classification of finite simple strongly real groups is complete. It is easy to see that strong reality for every nonabelian finite simple group is equivalent to the fact that each element can be written as a product of two involutions. We thus obtain a solution to Problem 14.82 of the Kourovka Notebook from the classification of finite simple strongly real groups. [ABSTRACT FROM AUTHOR]

Abstract: There are well-known monomorphisms between the Artin groups of finite type , and affine type , . The Artin group is isomorphic to the -strand braid group , and the other three Artin groups are isomorphic to some subgroups of . The inclusions between these subgroups yield monomorphisms , and . There are another type of monomorphisms , and which are induced by isomorphisms between Artin groups of type B and centralizers of periodic braids. In this paper, we show that the monomorphisms , and induce injective functions on the set of conjugacy classes, and that none of the monomorphisms , and does so. [Copyright &y& Elsevier]

We show by a combination of theoretical argument and computer search that if a projective (75, 4, 12, 5) set in PG(3, 7) exists then its automorphism group must be trivial. This corresponds to the smallest open case of a coding problem posed by H. Ward in 1998, concerning the possible existence of an infinite family of projective two-weight codes meeting the Griesmer bound. [ABSTRACT FROM AUTHOR]

The Agnihotri—Woodward—Belkale polytope Δ (resp., the Klyachko cone [Figure not available: see fulltext.]) is the set of solutions of the multiplicative (resp., additive) Horn problem, i.e., the set of triples of spectra of special unitary (resp. traceless Hermitian) n × n matrices satisfying AB = C (resp. A + B = C). The set [Figure not available: see fulltext.] is the tangent cone of Δ at the origin. The group G = ℝ n ⊕ ℝ n acts naturally on Δ. In this note, we report on a computer calculation showing that Δ coincides with the intersection of [Figure not available: see fulltext.], g ∈ G, for n ≤ 14 but does not coincide with it for n = 15. Our motivation was an attempt to understand how to solve the multiplicative Horn problem in practice for given conjugacy classes in SU( n). [ABSTRACT FROM AUTHOR]

Abstract: We prove that for any classical, compact, simple, connected Lie group G, the G-invariant orbital measures supported on non-trivial conjugacy classes satisfy a surprising -singular dichotomy: Either or is singular to the Haar measure on G. The minimum exponent k for which is specified; it depends on Lie properties of the element . As a corollary, we complete the solution to a classical problem – to determine the minimum exponent k such that for all central, continuous measures μ on G. Our approach to the singularity problem is geometric and involves studying the size of tangent spaces to the products of the conjugacy classes. [Copyright &y& Elsevier]

In this article, we show that for every abelian subgroup H of a Garside group, some conjugate g-1Hg consists of ultra summit elements and the centralizer of H is a finite index subgroup of the normalizer of H. Combining with the results on translation numbers in Garside groups, we obtain an easy proof of the algebraic flat torus theorem for Garside groups and solve several algorithmic problems concerning abelian subgroups of Garside groups. [ABSTRACT FROM AUTHOR]

Let G be a nonabelian group and associate a noncommuting graph ∇(G) with G as follows: The vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, Professor J. G. Thompson gave the following conjecture. Thompson's Conjecture. If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ≅ M, where N(G):={n ∈  | G has a conjugacy class of size n}. In 2006, A. Abdollahi, S. Akbari, and H. R. Maimani put forward a conjecture (AAM's conjecture) in Abdollahi et al. (2006) as follows. AAM's Conjecture. Let M be a finite nonabelian simple group and G a group such that ∇(G) ≅ ∇ (M). Then G ≅ M. In this short article we prove that if G is a finite group with ∇(G) ≅ ∇ (A10), then G ≅ A10, where A10 is the alternating group of degree 10. [ABSTRACT FROM AUTHOR]

The aim of this article is to give a generalization of the concept of commutativity degree of a finite group G (denoted by d(G)), to the concept of relative commutativity degree of a subgroup H of a group G (denoted by d(H, G)). We shall state some results concerning the new concept which are mostly new or improvements of known results given in Gustafson (1973) and Moghaddam et al. (2005). Moreover, we shall define the relative nth nilpotency degree of a subgroup of a group and give some results concerning this at the end of the article. [ABSTRACT FROM AUTHOR]

The aim of this note is to classify the finite solvable groups with an irreducible character that vanishes on just one class of elements. [ABSTRACT FROM AUTHOR]

We prove local finiteness for the groups generated by a conjugacy class of order 3 elements whose every pair generates a subgroup that is isomorphic to Z 3, A 4, A 5, SL 2(3), or SL 2(5). [ABSTRACT FROM AUTHOR]

Abstract: In a finite-dimensional algebra over a field F, a basis B is called a multiplicative basis provided that B ∪{0} forms a semigroup. We will describe all multiplicative bases of F n , the full algebra of n × n matrices over a subfield F of the real numbers. Every such basis is associated with a nonsingular zero–one matrix via a lattice order on F n. This association is a one-to-one correspondence after identification of isomorphic semigroups and identification of the zero–one matrices that have just permuted rows and columns. This correspondence yields an enumeration method for nonequivalent multiplicative bases of F n . The enumeration is done for n ⩽5. [Copyright &y& Elsevier]

It is known that if G is a generalized dihedral or quaternion 2-group, there exist two non-commuting elements x, y ∈ G such that x² = y². In this paper we generalize this result to all groups with a non-central cyclic normal subgroup of order 4. Some related results concerning groups and their real conjugacy classes are also obtained. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we define the class of finite groups of Suzuki type, which are non-abelian groups of exponent 4 and class 2 with special properties. A group G of Suzuki type with always possesses a non-trivial difference set. We show that if s is odd, G possesses a central difference set, whereas if s is even, G has no non-trivial central difference set. [Copyright &y& Elsevier]

Let k(G) be the number of conjugacy classes of finite groups G and πe(G) be the set of the orders of elements in G. Then there exists a non-negative integer k such that k(G) = |πe(G)| + k. We call such groups to be co(k) groups. This article classifies all finite co(1) groups. They are isomorphic to one of the following groups: A5, L2(7), S5, Z3, Z4, S5, A4, D10, Hol(Z5), or Z3 x Z4. [ABSTRACT FROM AUTHOR]

Let L≀X be a lamplighter graph, i.e., the graph-analogue of a wreath product of groups, and let P be the transition operator (matrix) of a random walk on that structure. We explain how methods developed by Saloff-Coste and the author can be applied for determining the ℓp-norms and spectral radii of P, if one has an amenable (not necessarily discrete or unimodular) locally compact group of isometries that acts transitively on L. This applies, in particular, to wreath products K≀G of finitely-generated groups, where K is amenable. As a special case, this comprises a result of Żuk regarding the ℓ2-spectral radius of symmetric random walks on such groups. [ABSTRACT FROM AUTHOR]

A subgroup of any group is called conjugately dense if it has nonempty intersection with each class of conjugate elements of the group. The aim of this paper is to prove the following. Let K be a locally finite field and H be an irreducible conjugately dense subgroup of the intermediate group SL3(K) ≤ G ≤ GL3(K); then H = G. This result confirms part of P. Neumann's conjecture from problem 6.38 in "Kourovka Notebook" for the group GL3(K) over locally finite field K. [ABSTRACT FROM AUTHOR]

Abstract: We construct a generator system of the annihilator of the generalized Verma module of induced from any character of any parabolic subalgebra as an analogue of minors and elementary divisors. The generator system has a quantization parameter and it generates the defining ideal of the conjugacy class of square matrices at the classical limit . [Copyright &y& Elsevier]

Let bcl(G) denote the largest conjugacy class length of a finite group G. In this note, we prove that if bcl(G)

Something between an expository note and an extended research problem, this article is an invitation to expand the existing literature on a family of graph invariants rooted in linear and multilinear algebra. There are a variety of ways to assign a real n×n matrix K(G) to each n-vertex graph G, so that G and H are isomorphic if and only if K(G) and K(H) are permutation similar. It follows that G and H are isomorphic only if K(G) and K(H) are similar, i.e., that similarity invariants of K(G) are graph theoretic invariants of G, an observation that helps to explain the enormous literature on spectral graph theory. The focus of this article is the permutation part, i.e., on matrix functions that are preserved under permutation similarity if not under all similarity. [Copyright &y& Elsevier]

Neumann characterized the groups in which every subgroup has finitely many conjugates only as central-by-finite groups. If [Xfr] is a class of groups, a group G is said to have [Xfr]-conjugate classes of subgroups if G/CoreG(NG(H)) ∈ [Xfr] for every subgroup H of G. In this paper, we generalize Neumann's result by showing that a group has polycyclic-by-finite classes of conjugate subgroup if and only if it is central-by-(polycyclic-by-finite). [ABSTRACT FROM AUTHOR]