Related papers: Finite basis problem for involution semigroups of …

Let $X$ be a finite set such that $|X|=n$, and let $k< n/2$. A group is $k$-homogeneous if it has only one orbit on the sets of size $k$. The aim of this paper is to prove some general results on permutation groups and then apply them to…

A finite semifield is a finite nonassociative ring with identity such that the set of its nonzero elements is closed under the product. From any finite semifield a projective plane can be constructed. In this paper we obtain new semifield…

We enumerate central and medial quasigroups of order less than $128$ up to isomorphism, with the exception of those quasigroups that are isotopic to $C_4\times C_2^4$, $C_2^6$, $C_3^4$ or $C_5^3$. We give an explicit formula for the number…

Necessary and sufficient conditions for finite semihypergroups to be built from groups of the same order are established

Let $G$ be a permutation group on the finite set $\Omega$. We prove various results about partitions of $\Omega$ whose stabilizers have good properties. In particular, in every solvable permutation group there is a set-stabilizer whose…

The finite orbits of the braid group action on Stokes matrices are studied and are shown to be the orbits on ordered sets of reflections, generating finite groups. All invariants of a reflection arrangement are determined. Determination of…

A group in which every element commutes with its endomorphic images is called an $E$-group. If $p$ is a prime number, a $p$-group $G$ which is an $E$-group is called a $pE$-group. Every abelian group is obviously an $E$-group. We prove that…

In this paper, we characterize the monoid of endomorphisms of the semigroup of all oriented full transformations of a finite chain, as well as the monoid of endomorphisms of the semigroup of all oriented partial transformations and the…

In this paper we completely classify semifields of order $2^8=256$ containing a nucleus of order $2^4=16$. We introduce new invariants for semifields, and apply new computational techniques for calculating old invariants. Together these…

Subfactor standard invariants encode quantum symmetries. The small index subfactor classification program has been a rich source of interesting quantum symmetries. We give the complete classification of subfactor standard invariants to…

Given a finitely generated residually finite group $G$, the residual finiteness growth $\text{RF}_G: \mathbb{N} \to \mathbb{N}$ bounds the size of a finite group $Q$ needed to detect an element of norm at most $r$. More specifically, if…

In this paper we explore the structure and properties of C-groups. We define a C-group as a group $G$ with $rk(G) < rk(Z(G))$ (where $rk(G)$ is the minimal cardinal of a generating set for a group $G$). Using GAP (a group theory program)…

We provide an example of minimum size of a finite semigroup with $\mathcal{J}^{\ast}\neq\mathcal{D}^{\ast}$. We introduce the notion of starred stability and prove that every starred stable semigroup has…

The most developed aspect of the theory of finite semigroups is their classification in pseudovarieties. The main motivation for investigating such entities comes from their connection with the classification of regular languages via…

A cover of a finite non-cyclic group $G$ is a family $\mathcal{H}$ of proper subgroups of $G$ whose union equals $G$. A cover of $G$ is called minimal if it has minimal size, and irredundant if it does not properly contain any other cover.…

We use the SmallGroups Library to find the finite subgroups of U(3) of order smaller than 512 which possess a faithful three-dimensional irreducible representation. From the resulting list of groups we extract those groups that can not be…

An explicit form of first order PDE for invariants of binary form are found. By solving the equation a minimal generation set for a ring of invariants and theirs syzygies are calculated in the cases $n\leq 6$ and $n=8.$

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot\ldots\cdot(n_lg)$ where $g\in G$ and $n_1, \ldots, n_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be the minimum of…

The complex representation rings of finite groups are the fundamental class of fusion rings, categorified by the corresponding fusion categories of complex representations. The category of $\mathbb{Z}_+$-modules of finite rank over such a…

We study algebraic properties of the semigroup $\mathscr{O\!\!I\!}_n(L)$ of finite partial order isomorphisms of the rank $\leq n$ of an infinite linearly ordered set $(L,\leqslant)$. In particular we describe its idempotents, the natural…