Narrow Results

Let $m$ be a cube-free positive integer and let $p$ be a prime such that $p\nmid m$. In this paper, we find the number of conjugacy classes of completely reducible solvable cube-free subgroups in $\mathrm{\text{GL}}(2,q)$ of order $m$, where $q$ is a power of $p$.

Let 𝖥_r be a free group of rank r, 𝔽_q a finite field of order q, and let SL_n(𝔽_q) act on Hom(𝖥_r, SL_n(𝔽_q)) by conjugation. We describe a general algorithm to determine the cardinality of the set of orbits Hom(𝖥_r, SL_n(𝔽_q))/SL_n(𝔽_q). Our first main theorem is the implementation of this algorithm in the case n = 2. As an application, we determine the E-polynomial of the character variety Hom(𝖥_r, SL₂(ℂ))//SL₂(ℂ), and of its smooth and singular locus. Thus we determine the Euler characteristic of these spaces.

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.

Giving a condition for the amenability of groups, Rosenblatt and Willis first introduced the concept of configuration. From the beginning of the theory, the question whether the concept of configuration equivalence coincides with the concept of group isomorphism was posed. We negatively answer this question by introducing two non-isomorphic, solvable and hence amenable groups which are configuration equivalent. Also, we will prove this conjecture, due to Rosenblatt and Willis, whether the configuration equivalent groups include the free non-Abelian group of the same rank or not. We show that two-sided equivalent groups have same class numbers.

In this paper, we count the number of SL₂(F_q)-representations of torus knot group up to conjugacy. For the finite field F_q case, the counting methods are interested in its own right. Explicit formulae of the effective counting are given in this paper.

We show that Nichols algebras of most simple Yetter–Drinfeld modules over the projective symplectic linear group over a finite field, corresponding to unipotent orbits, have infinite dimension. We give a criterion to deal with unipotent classes of general finite simple groups of Lie type and apply it to regular classes in Chevalley and Steinberg groups.

Let 𝔛 be a class of groups. A group G is said to be minimal non-𝔛 if all proper subgroups of G are 𝔛-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].

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.

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.

We study the semigroup structure on the set $C(A)$ of conjugacy classes of left ideals of a finite-dimensional algebra $A$ over an algebraically closed field $\mathrm{\text{K}}$, equipped with the natural multiplication inherited from $A$, and the structure of the contracted semigroup algebra ${\mathrm{\text{K}}}_0[C(A)]$. It is shown that $C(A)$ has a finite chain of ideals with either nilpotent or completely $0$-simple factors with trivial maximal subgroups, so in particular it is locally finite. The ordinary quiver $Q_A$ of $A$ is proved to be a subquiver of $Q_{{\mathrm{\text{K}}}_0[C(A)]}$, if $C(A)$ is finite. Moreover, in this case, the structure of $C(A)$ determines, up to isomorphism, the structure of the algebra $A$ modulo its Jacobson radical. Combining these results we show that if the semigroup $C(A)$ is finite, then it determines the structure of any (not necessarily basic) triangular algebra $A$ which admits a normed presentation.

Let $G$ be a finite AC-group such that $[G:Z(G)]=2^{n}s$, where $n\ge 3$, $s\ge 1$ and $s$ is odd. We prove that if $G$ has $2^{n+1}$ centralizers of elements, then $s=3$, $n$ is an even integer, the set of the sizes of the conjugacy classes of $G$ is $\{1,2^n,3\cdot 2^{n-1}\}$ and $G/Z(G)$ is a Frobenius group whose Frobenius kernel is an elementary abelian $2$-group of order $2^n$ and the Frobenius complement is a group of order $3$.

Let $G$ be a finite group. In this paper, we say that $g\in G$ is an SM-vanishing element of $G$, if there exists a strongly monolithic character $\chi$ of $G$ such that $\chi (g)=0$. The conjugacy class of an SM-vanishing element of $G$ is called an SM-vanishing conjugacy class of $G$. Our purpose here is to prove that for determining some properties of the structure of the group $G$, it is enough to consider the same arithmetical conditions on the sizes of SM-vanishing conjugacy classes of $G$ instead of certain arithmetical conditions on the sizes of vanishing conjugacy classes of $G$.

A constructive procedure is given to determine all ideals of a finite-dimensional solvable Lie algebra. This is used in determining all conjugacy classes of subalgebras of a given finite-dimensional solvable Lie algebra.

Let bcl(G) denote the largest conjugacy class length of a finite group G. In this note, we prove that if bcl(G)<p² for a prime p, then |G:O_p(G)|_p≤p.

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.

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.

Finite nilpotent groups having exactly four conjugacy classes of non-normal subgroups are classified.

In this paper we study the action of the fundamental group of a finite metric graph on its universal covering tree. We assume the graph is finite, connected and the degree of each vertex is at least three. Further, we assume an irrationality condition on the edge lengths. We obtain an asymptotic for the number of elements in a fixed conjugacy class for which the associated displacement of a given base vertex in the universal covering tree is at most T. Under a mild extra assumption we also obtain a polynomial error term.

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.

By comparing the influence of the order |G| of finite groups G to the structure of G, the influence of the set π_e(G) of all orders of elements in G to groups G is researched. In this article, the relative results are surveyed, and some unsolved problems are posed.