点连通度和边连通度的群论推广
Group-theoretic generalisations of vertex and edge connectivities
摘要：
设$p$是一个奇素数。设$P$是类为$2$，指数为$p$的有限$p$ -群，其交换子商$P/[P,P]$的阶为$p^n$ .我们定义了两个参数$P$相关的中心分解。第一个参数$κ(P)$是$P$的子群$S$存在的最小整数$s$，满足（1）$S\cap [P,P]=[S,S]$，（2）$|S/[S,S]|=p^{n-s}$，和（3）$S$允许非平凡中心分解。第二个参数，$λ(P)$，是最小的整数$s$的存在一个中心子群$P/N$的顺序$p^s$，使$P/N$允许一个非平凡的中心分解。虽然在纯群论术语中定义，但这两个参数分别概括了图的顶点连通度和边连通度：对于简单无向图$G$，通过Baer（Trans. Am.数学会，1938），Tutte（J. Lond.数学会，1947）和Lovász（B.布拉兹数学会，1989），存在一个类$2$和指数$p$ $P_G$的$p$ -群，它自然地与$G$相关联。我们的主要结果表明，点连通度$κ(G)$等于$κ(P_G)$，边连通度$λ(G)$等于$λ(P_G)$ .我们还讨论了一般$p$-类$2$和指数$p$的群$P$的$κ(P)$和$λ(P)$之间的关系，以及这些参数的计算方面。
英文摘要：
Let $p$ be an odd prime. Let $P$ be a finite $p$-group of class $2$ and exponent $p$, whose commutator quotient $P/[P,P]$ is of order $p^n$. We define two parameters for $P$ related to central decompositions. The first parameter, $κ(P)$, is the smallest integer $s$ for the existence of a subgroup $S$ of $P$ satisfying (1) $S\cap [P,P]=[S,S]$, (2) $|S/[S,S]|=p^{n-s}$, and (3) $S$ admits a non-trivial central decomposition. The second parameter, $λ(P)$, is the smallest integer $s$ for the existence of a central subgroup $N$ of order $p^s$, such that $P/N$ admits a non-trivial central decomposition. While defined in purely group-theoretic terms, these two parameters generalise respectively the vertex and edge connectivities of graphs: For a simple undirected graph $G$, through the classical procedures of Baer (Trans. Am. Math. Soc., 1938), Tutte (J. Lond. Math. Soc., 1947) and Lovász (B. Braz. Math. Soc., 1989), there is a $p$-group of class $2$ and exponent $p$ $P_G$ that is naturally associated with $G$. Our main results show that the vertex connectivity $κ(G)$ is equal to $κ(P_G)$, and the edge connectivity $λ(G)$ is equal to $λ(P_G)$. We also discuss the relation between $κ(P)$ and $λ(P)$ for a general $p$-group $P$ of class $2$ and exponent $p$, as well as the computational aspects of these parameters.
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