Möbius-Kantor图的半对称基本阿贝尔覆盖
Semisymmetric elementary abelian covers of the Möbius-Kantor graph
摘要：
设$\p_N \colon \tX \to X$是连通图的正则覆盖投影，其覆盖变换群同构于$\p_N \colon \tX \to X$.如果$N$是初等阿贝尔$p$ -群，则投影$\p_N$称为$p$ -初等阿贝尔群。如果$\Aut X$的某个顶点传递（边传递）子群沿$\p_N$提升，则投影$\p_N$是点传递（边传递）的;如果投影$\p_N$是边传递但不是点传递的，则投影$\p_N$是半对称的。投影$\p_N$是最小半对称的，如果$p_N$不能写成两个（非平凡的）正则覆盖投影的合成$\p_N = \p \circ \p_M$，其中$\p_M$是半对称的。寻找初等阿贝尔覆盖投影可以通过作用于图的第一同调群的自同构的线性表示来组合地把握。该方法本质上简化为寻找素域上矩阵群的不变子空间（参见{ \em J. Algebr. Combin.}，{ \bf 20}（2004），71- 97）。本文构造了Möbius-Kantor图的所有两两非同构的极小半对称初等阿贝尔正则覆盖投影，即广义Petersen图$\GP(8,3)$. $p =2$不存在此类封面。否则，这种覆盖投影的数目在情况$p \equiv 5,9,13,17,21 (\mod 24)$和$p \equiv 1 (\mod 24)$中分别等于$(p-1)/4$和$1+ (p-1)/4$，并且在情况$p \equiv 3,7,11,15,23 (\mod 24)$和$p \equiv 19 (\mod 24)$中分别等于$(p+1)/4$和$1+ (p+1)/4$。对于每个这样的覆盖投影，显式地显示生成对应覆盖的电压规则。
英文摘要：
Let $\p_N \colon \tX \to X$ be a regular covering projection of connected graphs with the group of covering transformations isomorphic to $N$. If $N$ is an elementary abelian $p$-group, then the projection $\p_N$ is called $p$-elementary abelian. The projection $\p_N$ is vertex-transitive (edge-transitive) if some vertex-transitive (edge-transitive) subgroup of $\Aut X$ lifts along $\p_N$, and semisymmetric if it is edge- but not vertex-transitive. The projection $\p_N$ is minimal semisymmetric if $p_N$ cannot be written as a composition $\p_N = \p \circ \p_M$ of two (nontrivial) regular covering projections, where $\p_M$ is semisymmetric. Finding elementary abelian covering projections can be grasped combinatorially via a linear representation of automorphisms acting on the first homology group of the graph. The method essentially reduces to finding invariant subspaces of matrix groups over prime fields (see {\em J. Algebr. Combin.}, {\bf 20} (2004), 71--97). In this paper, all pairwise nonisomorphic minimal semisymmetric elementary abelian regular covering projections of the Möbius-Kantor graph, the Generalized Petersen graph $\GP(8,3)$, are constructed. No such covers exist for $p =2$. Otherwise, the number of such covering projections is equal to $(p-1)/4$ and $1+ (p-1)/4$ in cases $p \equiv 5,9,13,17,21 (\mod 24)$ and $p \equiv 1 (\mod 24)$, respectively, and to $(p+1)/4$ and $1+ (p+1)/4$ in cases $p \equiv 3,7,11,15,23 (\mod 24)$ and $p \equiv 19 (\mod 24)$, respectively. For each such covering projection the voltage rules generating the corresponding covers are displayed explicitly.
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