PreprintCC BY 4.0arXiv
On the strong domination relation of order sequences for some group extensions
Abstract
Given a finite group G, its order sequence os(G) is a sequence formed of the element orders of G written in non-decreasing order. In [Electron. J. Combin. 32 (2025), no. 2, Paper No. 2.9], P. J. Cameron and H. K. Dey proved that if G and H are groups of coprime orders, G is abelian and K is any extension of G by H, then there is a bijection f from G× H to K such that the order of f(x) divides the order of x, for all x∈ G× H. In the same paper, the authors asked if the hypothesis ``G is abelian" is necessary. We show that if this condition is dropped, the result does not hold in general.
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BibTeX
@misc{lazorec2026strong,
title = {{On the strong domination relation of order sequences for some group extensions}},
author = {Mihai-Silviu Lazorec},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24950},
eprint = {2610.24950},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24950},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the strong domination relation of order sequences for some group extensions
AU - Mihai-Silviu Lazorec
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.24950
UR - https://arxiv.org/abs/2610.24950
AB - Given a finite group G, its order sequence os(G) is a sequence formed of the element orders of G written in non-decreasing order. In [Electron. J. Combin. 32 (2025), no. 2, Paper No. 2.9], P. J. Cameron and H. K. Dey proved that if G and H are groups of coprime orders, G is abelian and K is any extension of G by H, then there is a bijection f from G× H to K such that the order of f(x) divides the order of x, for all x∈ G× H. In the same paper, the authors asked if the hypothesis ``G is abelian" is necessary. We show that if this condition is dropped, the result does not hold in general.
ER -