Algebraic elements in group rings

Lie properties of symmetric elements in group rings

Let R be a non trivial finite commutative ring with identity and G be a non trivial group. We denote by P(RG) the probability that the product of two randomly chosen elements of a finite group ring RG is zero. We show that P(RG) <0.25 if and only if RG is not isomorphic to Z2C2, Z3C2, Z2C3. Furthermore, we give the upper bound and lower bound for P(RG). In particular, we present the general formula for P(RG), where R is a finite field of characteristic p and |G| ≤ 4.

The aim of this article is to contribute to a question of R. Brauer that “when do non-isomorphic groups have isomorphic complex group algebras?” Let H and G be finite groups where and let denote the first column of the complex character table of H. In this article, we show that if then provided that q + 1 divides neither n nor n – 1. Consequently, it is shown that G is uniquely determined by the structure of its complex group algebra. This in particular extends a recent result of Bessenrodt et al. [Algebra Number Theory 9 (2015), 601–628] to the almost simple groups of arbitrary rank.

We prove that there are examples of finitely generated groups G together with group ring elements Q \in \bbQ G for which the von Neumann dimension \dim_{LG}\ker Q is irrational, so (in conjunction with other known results) answering a question of Atiyah.

Let \(G\) be a finite group. The character degree graph of \(G\), which is denoted by \(\Gamma (G)\), is the graph whose vertices are the prime divisors of the character degrees of the group \(G\) and two vertices \(p_1\) and \(p_2\) are joined by an edge if \(p_1p_2\) divides some character degree of \(G\). In this paper we prove that the simple group \(\mathrm{PSL}(2,p^2) \) is uniquely determined by its character degree graph and its order. Let \(X_1(G)\) be the set of all irreducible complex character degrees of \(G\) counting multiplicities. As a consequence of our results we prove that if \(G\) is a finite group such that \(X_1(G)=X_1(\mathrm{PSL}(2,p^2) )\), then \(G\cong \mathrm{PSL}(2,p^2) \). This implies that \(\mathrm{PSL}(2,p^2) \) is uniquely determined by the structure of its complex group algebra.

Almost simple groups with the socle $PSL(2,p^n)$ are determined by their complex group algebras

In this paper, we determine all of finite groups whose order and the largest of their irreducible character degrees are the same as $${\textit{PGL}}( 2 , p^{2} ) $$ for all odd prime numbers p. As a consequence, we show that the groups $${\textit{PGL}}( 2 , p^{2} ) $$ are uniquely determined by the structure of their complex group algebras.

Simple classical groups of Lie type are determined by their character degrees

Exact algebraic algorithms for calculating the product of two elements of nilpotent associative algebras over fields of characteristic zero are considered (this is a particular case of simultaneous calculation of several multinomials). The complexity of an algebra in this computational model is defined as the number of nonscalar multiplications of an optimal algorithm. Lower bounds for the tensor rank of nilpotent associative algebras (in terms of dimensions of certain subalgebras) are obtained, which give lower bounds for the algebraic complexity of this class of algebras. Examples of reaching these estimates for different dimensions of nilpotent algebras are presented.

In this paper, we employ group rings and some known results on group codes to study reversible group DNA codes. We define and study reversible cyclic DNA codes from a group ring point of view and we also introduce the notion for self-reciprocal group ring elements. Moreover, we search for reversible group DNA codes with the use of a virus optimisation algorithm. We obtain many good DNA codes that satisfy the Hamming distance, the reverse, the reverse-complement and the fixed GC-content constraints.

Mutually unbiased bases (MUBs) are important in quantum information theory. While constructions of complete sets of d + 1 MUBs in Copf <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> are known when d is a prime power, it is unknown if such complete sets exist in non-prime power dimensions. It has been conjectured that sets of complete MUBs only exist in Copf <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> if a projective plane of size d also exists. We investigate the structure of MUBs using two algebraic tools: relation algebras and group rings. We construct two relation algebras from MUBs and compare these to relation algebras constructed from projective planes. We show several examples of complete sets of MUBs in Copf <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> , that when considered as elements of a group ring form a commutative monoid. We conjecture that complete sets of MUBs will always form a monoid if the appropriate group ring is chosen.

Some primitive group rings

Let $G$ be a complex algebraic group and $L$ an algebraic subgroup of $G$. The quotient space $G/L$ is called weakly commutative if a generic orbit of the action $G:T^*(G/L)$ is a coisotropic submanifold. We classify weakly commutative homogeneous spaces $N\leftthreetimes L/L$ in the case where $L$ is a reductive group and the natural representation $L:\mathfrak n/[\mathfrak n,\mathfrak n]$, where $\mathfrak n$ is the tangent algebra of the group $N$, is irreducible.

A lax monoidal topological quantum field theory for representation varieties

Associative rings with metabelian adjoint group