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On the complement of a union of cosets
Abstract
Let g1H1,…,gnHn 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 2n left translates of A:=G∖ U. For finite G this yields |A|≥|G|/2n, which settles a conjecture of Tărnăuceanu and the author. Equality |A|=|G|/2n can only hold when A is a left coset of K:=H1∩…∩ Hn. Moreover, every subgroup occurring in an irredundant cover of an arbitrary group by n cosets has index at most 2n−1. This improves a lemma of Neumann.
Related
BibTeX
@misc{sambale2026complement,
title = {{On the complement of a union of cosets}},
author = {Benjamin Sambale},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09052},
eprint = {2609.09052},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09052},
}
RIS
TY - UNPB
TI - On the complement of a union of cosets
AU - Benjamin Sambale
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09052
UR - https://arxiv.org/abs/2609.09052
AB - Let g_1H_1,…,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∖ U. For finite G this yields |A|≥|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∩…∩ 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.
ER -