Year: 2026
Type: preprint
Abstract: Let $g_1H_1,\ldots,g_nH_n$ be left cosets of subgroups of a group $G$ whose union $U$ is a proper subset of $G$. We prove that $G$ is the union of at most $2^n$ left translates of $A:=G\setminus U$. For finite $G$ this yields $|A|\ge|G|/2^n$, which settles a conjecture of Tărnăuceanu and the author. Equality $|A|=|G|/2^n$ can only hold when $A$ is a left coset of $K:=H_1\cap\ldots\cap H_n$. Moreover, every subgroup occurring in an irredundant cover of an arbitrary group by $n$ cosets has index at most $2^{n-1}$. This improves a lemma of Neumann.
Source: arXiv (Cornell University)
Author: Sambale, Benjamin
Institution: Leibniz University Hannover
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212037224