PreprintCC BY 4.0arXiv
An observation on factorizations of finite groups
Abstract
Given a finite group G and a factorization of its order, |G|= a1 a2, we show that a sufficient condition for there to exist subsets A1, A2⊆ G such that |A1| = a1, |A2| = a2, and G = A1 A2, is that there exist a chain of subgroups {e} = G0 < … < Gn = G such that a1 is the product of some subfamily of the indices |Gi:Gi−1| (and hence a2 is the product of the complementary subfamily).
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BibTeX
@misc{bergman2026observation,
title = {{An observation on factorizations of finite groups}},
author = {George M. Bergman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.08928},
eprint = {2608.08928},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.08928},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An observation on factorizations of finite groups
AU - George M. Bergman
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2608.08928
UR - https://arxiv.org/abs/2608.08928
AB - Given a finite group G and a factorization of its order, |G|= a_1 a_2, we show that a sufficient condition for there to exist subsets A_1, A_2⊆ G such that |A_1| = a_1, |A_2| = a_2, and G = A_1 A_2, is that there exist a chain of subgroups {e} = G_0 < … < G_n = G such that a_1 is the product of some subfamily of the indices |G_i:G_i−1| (and hence a_2 is the product of the complementary subfamily).
ER -