Let $S,T$ be two distinct finite Abelian groups with $|S|=|T|$. A fundamental theorem of Tutte shows that a graph admits a nowhere-zero $S$-flow if and only if it admits a nowhere-zero $T$-flow. Jaeger, Linial, Payan and Tarsi in 1992 introduced group connectivity as an extension of flow theory, and they asked whether such a relation holds for group connectivity analogy. It was negatively answered by Husek, Mohelnikova and Samal in 2017 for graphs with edge-connectivity 2 for the groups $S=\mathbb{Z}_4$ and $T=\mathbb{Z}_2^2$. In this paper, we extend their results to $3$-edge-connected graphs (including both cubic and general graphs), which answers open problems proposed by Husek, Mohelnikova and Samal(2017) and Lai, Li, Shao and Zhan(2011). Combining some previous results, this characterizes all the equivalence of group connectivity under $3$-edge-connectivity, showing that every $3$-edge-connected $S$-connected graph is $T$-connected if and only if $\{S,T\}\neq \{\mathbb{Z}_4,\mathbb{Z}_2^2\}$.

中文翻译：

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三边连接下的群连接

令 $S,T$ 是两个不同的有限阿贝尔群，其中 $|S|=|T|$。Tutte 的一个基本定理表明，当且仅当它承认一个无处为零的 $T$ 流时，图才允许无处为零的 $S$ 流。Jaeger、Linial、Payan 和 Tarsi 在 1992 年引入了群连通性作为流动理论的扩展，他们询问这种关系是否适用于群连通性类比。Husek、Mohelnikova 和 Samal 在 2017 年对 $S=\mathbb{Z}_4$ 和 $T=\mathbb{Z}_2^2$ 组的边连通性为 2 的图给出了否定的回答。在本文中，我们将他们的结果扩展到 $3$-边连接图（包括三次图和一般图），它回答了 Husek、Mohelnikova 和 Samal（2017）以及 Lai、Li、Shao 和 Zhan（2011）提出的开放问题. 结合之前的一些结果，