A note on Closure Operators in Category of Groups

$E$-酉幺半群与$F$-逆幺半群，以及群Cayley图上的闭包算子

$E$-unitary and $F$-inverse monoids, and closure operators on group Cayley graphs

本文建立了$E$-酉逆幺半群与$F$-逆幺半群分别等价于群Cayley图上特定闭包算子范畴，并引入$F$-Schützenberger图与$P$-展开以研究其表示。

A note on a free group. The decomposition of a free group functor through the category of heaps

This note aims to introduce a left adjoint functor to the functor which assigns a heap to a group. The adjunction is monadic. It is explained how one can decompose a free group functor through the previously introduced adjoint and employ it to describe a slightly different construction of free groups.

A Note on Parabolic Subgroups of a Coxeter Group

The aim of this note is to prove that the parabolic closure of any subset of a Coxeter group is a parabolic subgroup. To obtain that, several technical lemmas on the root system of a parabolic subgroup are established.

A note on affine links

We view the $\tilde{A}$-type affine braid group as a subgroup of the $B$-type braid group. We show that the $\tilde{A}$-type affine braid group surjects onto the $A$-type braid group and we detect the kernel of this surjection using Schreier's Theorem. We then describe an injection of the $B$-type braid group into the $A$-type braid group which allows us finally to give a definition of affine links, as closures of affine braids viewed as A-type braids after composing the above injections, and we prove that the two conditions of Markov are necessary and sufficient to get the same affine closure…

Notes on finite totally $2$-closed permutation groups

Let $N$ be a normal subgroup of a finite group $G$. For a faithful $N$-set $Δ$, applying the university embedding theorem one can construct a faithful $G$-set $Ω$. In this short note, it is proved that if the $2$-closure of $N$ in $Ω$ is equal to $N$, then the $2$-closure of $N$ in $Δ$ is also equal to $N$; in addition, it is proved that any abelian normal subgroup of a finite totally $2$-closed group is cyclic; finally, it is proved that if a finite nilpotent group is a direct of two nilpotent subgroups where the two factors have coprime orders and both of them are totally 2-closed then G is …

Cocharacter-closure and spherical buildings

Let $k$ be a field, let $G$ be a reductive $k$-group and $V$ an affine $k$-variety on which $G$ acts. In this note we continue our study of the notion of cocharacter-closed $G(k)$-orbits in $V$. In earlier work we used a rationality condition on the point stabilizer of a $G$-orbit to prove Galois ascent/descent and Levi ascent/descent results concerning cocharacter-closure for the corresponding $G(k)$-orbit in $V$. In the present paper we employ building-theoretic techniques to derive analogous results.