International Journal of Group Theory
Volume:12 Issue: 3, Sep 2023

We consider Kazhdan-Lusztig cells of the symmetric group $S_n$ containing the longest element of a standard parabolic subgroup of $S_n$. Extending some of the ideas in [Beiträge zur Algebra und Geometrie, 59 (2018) no.~3 523--547] and [Journal of Algebra and Its Applications, 20 (2021) no.~10 2150181], we determine the rim of some additional families of cells and also of certain induced unions of cells. These rims provide minimal determining sets for certain $W$-graph ideals introduced in [Journal of Algebra, 361 (2012) 188--212].

A rather natural way for trying to obtain a lattice-theoretic characterization of a class of groups ${\mathcal X}$ is to replace the concepts appearing in the definition of ${\mathcal X}$ by lattice-theoretic concepts. The first to use this idea were Kontorovi\v{c} and Plotkin who in 1954 introduced the notion of modular chain in a lattice, as translation of a central series of a group, to determine a lattice-theoretic characterization of the class of torsion-free nilpotent groups. The aim of this paper is to present a recent application of this translation method to some generalized nilpotency properties.

Let $\mathcal{M}$ be a family of maximal subgroups of a group $G.$ We say that $\mathcal{M}$ is irredundant if its intersection is not equal to the intersection of any proper subfamily of $\mathcal{M}$. The maximal dimension of $G$ is the maximal size of an irredundant family of maximal subgroups of $G$. In this paper we study a class of solvable groups, called $\mathcal{M}$-groups, in which the maximal dimension has properties analogous to that of the dimension of a vector space such as the span property, the extension property and the basis exchange property.

We survey the known group structures arising from elliptic curves defined by Weierstrass models over commutative rings with unity and satisfying a technical condition. For every considered base ring, the groups that may arise depending on the curve coefficients are recalled. When a complete classification is still out of reach, partial results about the group structure and relevant subgroups are provided. Several examples of elliptic curves over the inspected rings are presented, and open questions regarding the structure of their points are highlighted.

We extend the definition of boundary expansion to CW complexes and prove a Cheeger-Buser-type relation between the spectral gap of the Laplacian and the boundary expansion of an orientable CW complex.

For a semigroup $S$, the covering number of $S$ with respect to semigroups, $\sigma_s(S)$, is the minimum number of proper subsemigroups of $S$ whose union is $S$. This article investigates covering numbers of semigroups and analogously defined covering numbers of inverse semigroups and monoids. Our three main theorems give a complete description of the covering number of finite semigroups, finite inverse semigroups, and monoids (modulo groups and infinite semigroups). For a finite semigroup that is neither monogenic nor a group, its covering number is two. For all $n\geq 2$, there exists an inverse semigroup with covering number $n$, similar to the case of loops. Finally, a monoid that is neither a group nor a semigroup with an identity adjoined has covering number two as well.