Harmonious sequences in groups with a unique involution

Mohammad Javaheri, Lydia de Wolf
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Abstract

We study several combinatorial properties of finite groups that are related to the notions of sequenceability, R-sequenceability, and harmonious sequences. In particular, we show that in every abelian group $G$ with a unique involution $\imath_G$ there exists a permutation $g_0,\ldots, g_{m}$ of elements of $G \backslash \{\imath_G\}$ such that the consecutive sums $g_0+g_1, g_1+g_2,\ldots, g_{m}+g_0$ also form a permutation of elements of $G\backslash \{\imath_G\}$. We also show that in every abelian group of order at least 4 there exists a sequence containing each non-identity element of $G$ exactly twice such that the consecutive sums also contain each non-identity element of $G$ twice. We apply several results to the existence of transversals in Latin squares.
具有独特内卷的群体中的和谐序列
我们研究了有限群的几个组合性质,这些性质与可序列性、R-可序列性和和谐序列等概念有关。特别是,我们证明了在每一个具有唯一内卷$/imath_G$的无性群$G$中,都存在一个$g_0,\ldots、的元素的排列组合 $g_{m}$,使得连续和 $g_0+g_1,g_1+g_2,\ldots, g_{m}+g_0$ 也构成 $G\backslash\{imath_G\}$ 的元素的排列组合。我们还证明,在每一个阶数至少为 4 的无性群中,都存在一个包含 $G$ 的每个非同位元素两次的序列,这样连续的和也包含 $G$ 的每个非同位元素两次。我们将几个结果应用于拉丁方阵中横轴的存在性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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