arXiv subjects

Luise-Charlotte Kappe

Publications and source records attributed to Luise-Charlotte Kappe.

3 recordsLinked to original sources

arXiv2020

Finite Coverings of Semigroups and Related Structures

For a semigroup $S$, the covering number of $S$ with respect to semigroups, $σ_s(S)$, 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.

math.GR↗

arXiv2018

On integers that are covering numbers of groups

The covering number of a group $G$, denoted by $σ(G)$, is the size of a minimal collection of proper subgroups of $G$ whose union is $G$. We investigate which integers are covering numbers of groups. We determine which integers $129$ or smaller are covering numbers, and we determine precisely or bound the covering number of every primitive monolithic group with a degree of primitivity at most $129$ by introducing effective new computational techniques. Furthermore, we prove that, if $\mathscr{F}_1$ is the family of finite groups $G$ such that all proper quotients of $G$ are solvable, then $\mathbb{N}-\{σ(G):G\in \mathscr{F}_1\}$ is infinite, which provides further evidence that infinitely many integers are not covering numbers. Finally, we prove that every integer of the form $(q^m-1)/(q-1)$, where $m\neq3$ and $q$ is a prime power, is a covering number, generalizing a result of Cohn.

math.GR↗

arXiv2014

On the Covering Number of Small Symmetric Groups and Some Sporadic Simple Groups

A set of proper subgroups is a covering for a group if its union is the whole group. The minimal number of subgroups needed to cover $G$ is called the covering number of $G$, denoted by $σ(G)$. Determining $σ(G)$ is an open problem for many non-solvable groups. For symmetric groups $S_n$, Maróti determined $σ(S_n)$ for odd $n$ with the exception of $n=9$ and gave estimates for $n$ even. In this paper we determine $σ(S_n)$ for $n = 8$, $9$, $10$ and $12$. In addition we find the covering number for the Mathieu group $M_{12}$ and improve an estimate given by Holmes for the Janko group $J_1$.

math.GR↗