The special subgroup of invertible non-commutative rational power series as a metric group

Invertible and nilpotent matrices over antirings

In this paper we characterize invertible matrices over an arbitrary commutative antiring S and find the structure of GL_n (S). We find the number of nilpotent matrices over an entire commutative finite antiring. We prove that every nilpotent $n \times n$ matrix over an entire antiring can be written as a sum of $\lceil \log_2 n \rceil$ square-zero matrices and also find the necessary number of square-zero summands for an arbitrary trace-zero matrix to be expressible as their sum.

Non-commutative lattice problems

We consider several subgroup-related algorithmic questions in groups, modeled after the classic computational lattice problems, and study their computational complexity. We find polynomial time solutions to problems like finding a subgroup element closest to a given group element, or finding a shortest non-trivial element of a subgroup in the case of nilpotent groups, and a large class of surface groups and Coxeter groups. We also provide polynomial time algorithm to compute geodesics in given generators of a subgroup of a free group.

非交换纠错码与真子群检测

Non-commutative error correcting codes and proper subgroup testing

本文研究通过随机元素检测子群是否为真子群的问题，将其归结为非交换纠错码的构造，证明了线性大小且具有常数检测概率的测试子集存在，从而推广了Shannon经典纠错码结果，并在多类群中给出具体构造。

A new look at the Burnside-Schur theorem

The famous Burnside-Schur theorem states that every primitive finite permutation group containing a regular cyclic subgroup is either 2-transitive or isomorphic to a subgroup of a 1-dimensional affine group of prime degree. It is known that this theorem can be expressed as a statement on Schur rings over a finite cyclic group. Generalizing the latters we introduce Schur rings over a finite commutative ring and prove an analog of this statement for them. Besides, the finite local commutative rings are characterized in the permutation group terms.

Normal cyclotomic schemes over a finite commutative ring

We study cyclotomic association schemes over a finite commutative ring $R$ with identity. The main interest for us is to identify the normal cyclotomic schemes $C$, i.e. those for which $Aut(C)$ is a subgroup of the one-dimensional affine semilinear group over $R$. The problem is reduced to the case when the ring $R$ is local in which a necessary condition of normality in terms of the subgroup of $R^\times$ defining $C$, is given. This condition is proved to be sufficient for a class of local rings including the Galois rings of odd characteristic.

${\bf {\rm Sp}(n,2)}$ 上的 Schur 环与重数为一的子群

Schur rings over ${\bf {\rm Sp}(n,2)}$ and multiplicity one subgroups

本研究针对辛群Sp(n,2)上含辛平延类的交换Schur环展开，确定了该类的可行划分，明确了其对Sp(n,2)重数为一子群可能性的约束作用。