We establish a link between abelian regular subgroup of the affine group, and commutative, associative algebra structures on the underlying vector space that are (Jacobson) radical rings. As an application, we show that if the underlying field has positive characteristic , then an abelian regular subgroup has finite exponent if the vector space is finite-dimensional, while it can be torsion fre...

the non-commuting graph $nabla(g)$ of a non-abelian group $g$ is defined as follows: its vertex set is $g-z(g)$ and two distinct vertices $x$ and $y$ are joined by an edge if and only if the commutator of $x$ and $y$ is not the identity. in this paper we 'll prove that if $g$ is a finite group with $nabla(g)congnabla(bs_{n})$, then $g cong bs_{n}$, where $bs_{n}$ is the symmetric group of degre...

We establish a correspondence between abelian regular subgroup of the affine group, and commutative, associative algebra structures on the underlying vector space that are (Jacobson) radical rings. As an application, we show that if the underlying field has positive characteristic, then an abelian regular subgroup has finite exponent if the vector space is finite-dimensional, while it can be to...

In this note we characterize thick finite generalized quadrangles constructed from a generalized hyperoval as those admitting an abelian Singer group, i.e., an abelian group acting regularly on the points.

We obtain an estimate of Voiculescu’s (modified) free entropy dimension for generators of a II1-factor M with a subfactor N containing an abelian subalgebra A of finite multiplicity. It implies in particular that the interpolated free group subfactors of finite Jones index do not have abelian subalgebras of finite multiplicity or Cartan subalgebras.

1. The automorphism group of any group G we denote by Y(G). We shall consider the following two types of subgroups of T(G) : if H is a subgroup of G, Y(G: H) is the subgroup consisting of all automorphisms which are the identity on H, and if H is a normal subgroup, Y(G: G/H) is the subgroup consisting of all automorphisms which leave H invariant and which induce the identity on G/H. For the spe...

This article is motivated by the following local-to-global question: is every variety with tame quotient singularities globally the quotient of a smooth variety by a finite group? We show that this question has a positive answer for all quasi-projective varieties which are expressible as a quotient of a smooth variety by a split torus (e.g. simplicial toric varieties). Although simplicial toric...

One of the main aims of workers in the theory of groups has always been the determination of all finite simple groups. For simple groups may be regarded as the fundamental building blocks out of which finite groups are constructed. The cyclic groups of prime order are trivial examples of simple groups, and are the only simple groups which are Abelian. The first examples of non-Abelian simple gr...

We prove a general form of bit flip formula for the quantum Fourier transform on finite abelian groups and use it to encode some general CSS codes on these groups.

A topological group G is said to be almost maximally almost-periodic if its von Neumann radical n(G) is non-trivial, but finite. In this paper, we prove that (a) every countably infinite abelian torsion group, (b) every abelian torsion group of cardinality greater than continuum, and (c) every (non-trivial) divisible abelian torsion group admits a (Hausdorff) almost maximally almost-periodic gr...