Let $G$ be a finite additive abelian group with exponent $n>1$, and let $a_1,\ldots,a_{n-1}$ be elements of $G$. We show that there is a permutation $\sigma\in S_{n-1}$ such that all the elements $sa_{\sigma(s)}\ (s=1,\ldots,n-1)$ are nonzero if and only if$$\left|\left\{1\leqslant s<n:\ \frac{n}da_s\not=0\right\}\right|\geqslant d-1\ \ \mbox{for any positive divisor}\ d\ \mbox{of}\ n.$$When $G$ is the cyclic group $\mathbb Z/n\mathbb Z$, this confirms a conjecture of Z.-W. Sun.
@article{10_37236_5915,
author = {Fan Ge and Zhi-Wei Sun},
title = {On a permutation problem for finite abelian groups},
journal = {The electronic journal of combinatorics},
year = {2017},
volume = {24},
number = {1},
doi = {10.37236/5915},
zbl = {1355.05008},
url = {http://geodesic.mathdoc.fr/articles/10.37236/5915/}
}
TY - JOUR
AU - Fan Ge
AU - Zhi-Wei Sun
TI - On a permutation problem for finite abelian groups
JO - The electronic journal of combinatorics
PY - 2017
VL - 24
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/5915/
DO - 10.37236/5915
ID - 10_37236_5915
ER -
%0 Journal Article
%A Fan Ge
%A Zhi-Wei Sun
%T On a permutation problem for finite abelian groups
%J The electronic journal of combinatorics
%D 2017
%V 24
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/5915/
%R 10.37236/5915
%F 10_37236_5915
Fan Ge; Zhi-Wei Sun. On a permutation problem for finite abelian groups. The electronic journal of combinatorics, Tome 24 (2017) no. 1. doi: 10.37236/5915
