Related papers: On Generalized Commutator

We prove that, for every integer $n \ge 2$, a finite or infinite countable group $G$ can be embedded into a 2-generated group $H$ in such a way that the solvability of quadratic equations of length at most $n$ is preserved, i.e., every…

We study systematically groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth…

A geometric interpretation and generalisation for the Galois action on finite group character tables is sketched. The generalisation is a Galois action on the space Map_G(G^n,\bar{Q})/S_n for each finite G, where G acts by simultaneous…

A class of generalized nonlinear Kolmogorov equations is investigated. We present the group classification of Lie symmetries of the class with respect to the group of equivalence transformations. We find a number of exact solutions of…

Let $G$ be a finite group. For all $a \in \Z$, such that $(a,|G|)=1$, the function $\rho_a: G \to G$ sending $g$ to $g^a$ defines a permutation of the elements of $G$. Motivated by a recent generalization of Zolotarev's proof of classic…

Let G be a linear group such that for every g in G there is a finite set R(g) with the property that for every x in G all sufficiently long commutators [g,x,x,...,x] belong to R(g). It is proved that G is finite-by-hypercentral.

Let $\mathbb{F}_q$ be the finite field of $q$ elements and $a_1,a_2, \ldots, a_k, b\in \mathbb{F}_q$. We investigate $N_{\mathbb{F}_q}(a_1, a_2, \ldots,a_k;b)$, the number of ordered solutions $(x_1, x_2, \ldots,x_k)\in\mathbb{F}_q^k$ of…

We prove that the solvable radical of a finite group G coincides with the set of elements y having the following property: for any x in G the subgroup of G generated by x and y is solvable. We present analogues of this result for finite…

We present a characterisation of groups $G$ of order $p^7$, $p$ prime, in which not all elements of the commutator subgroup $\gamma_2(G)$ of $G$ are commutators in $G$. On the way we obtain several structural results on groups of order…

Let $G$ be a finite group and \( M \) be a maximal subgroup of \( G \). We call every irreducible constituent \( \chi \) of \( (1_M)^G \) a \( \mathcal{P} \)-character of \( G \) with respect to \( M \). In this paper, we prove that all…

Many invariants of finitely generated positive cancelative commutative semigroups can be studied from their Poincar\'e series. We offer and present several closed formulas for them. Moreover, those formulas have elementary proofs and are…

We describe the "generic" part of the character ring of general linear groups over a finite field in terms of quiver representations.

We show that the integral \int e^{S(x_1,...,x_n)}dx_1...dx_n, for an arbitrary polynomial S, satisfies a generalized hypergeometric system of differential equations in the sense of I. M. Gelfand et al.

Preface (A.Vershik) - about these texts (3.); I.Interpolation between inductive and projective limits of finite groups with applicatons to linear groups over finite fields; II.The characters of the groups of almost triangle matrices over…

We define new generalized factorials in several variables over an arbitrary subset $\underline{S} \subseteq R^n,$ where $R$ is a Dedekind domain and $n$ is a positive integer. We then study the properties of the fixed divisor…

In this paper, the changes of representations of a group are used in order to describe its action as algebraic Galois group of an univariate polynomial on the roots of factors of any Lagrange resolvent. By this way, the Galois group of…

We show that the following group constructions preserve the semilinearity of the solution sets for knapsack equations (equations of the form $g_1^{x_1} \cdots g_k^{x_k} = g$ in a group $G$, where the variables $x_1, \ldots, x_k$ take values…

The Burnside Problem asks whether a finitely generated group of exponent n is finite. We present a solution for 2-generator groups of prime power exponent. Results of P. Hall and G. Higman extends the finiteness conclusion to groups having…

In this paper we investigate congruence relationships of particular finite generalized harmonic numbers sums. We suggest more transparent and simpler method to analyse these sums and present several additional results for certain special…

Let A be a finite set of integers. For a polynomial f(x_1,...,x_n) with integer coefficients, let f(A) = {f(a_1,...,a_n) : a_1,...,a_n \in A}. In this paper it is proved that for every pair of normalized binary linear forms f(x,y)=u_1x+v_1y…