PreprintCC BY 4.0arXiv
A simultaneous variant of the Hall-Paige theorem and applications
Abstract
For k≥ 1 , we say that a finite group G satisfies property 𝒫k if, for every β1,…,βk∈Aut(G), there exists a bijection f of G such that each map g↦ βi(g)f(g) is a bijection. For k=1 this is equivalent to the classical Hall--Paige complete mapping property. We prove that if p is the smallest prime divisor of |G|, then G satisfies 𝒫k for every k<p−1 and we show that this bound is sharp. For finite abelian p-groups, the bijection f may moreover be chosen to be an automorphism, and for noncyclic homocyclic p-groups this remains true for k=p−1. Finally, we interpret 𝒫k in terms of common transversals and common orthogonal mates of Latin squares generated from automorphisms and the Cayley table of a finite group G.
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BibTeX
@misc{loukaki2026simultaneous,
title = {{A simultaneous variant of the Hall-Paige theorem and applications}},
author = {Maria Loukaki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21600},
eprint = {2610.21600},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21600},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A simultaneous variant of the Hall-Paige theorem and applications
AU - Maria Loukaki
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.21600
UR - https://arxiv.org/abs/2610.21600
AB - For k≥ 1 , we say that a finite group G satisfies property 𝒫_k if, for every β_1,…,β_k∈Aut(G), there exists a bijection f of G such that each map g↦ β_i(g)f(g) is a bijection. For k=1 this is equivalent to the classical Hall--Paige complete mapping property. We prove that if p is the smallest prime divisor of |G|, then G satisfies 𝒫_k for every k<p−1 and we show that this bound is sharp. For finite abelian p-groups, the bijection f may moreover be chosen to be an automorphism, and for noncyclic homocyclic p-groups this remains true for k=p−1. Finally, we interpret 𝒫_k in terms of common transversals and common orthogonal mates of Latin squares generated from automorphisms and the Cayley table of a finite group G.
ER -