# 阶为8p2的7度对称图On Symmetric Graphs of Order Eight Times a Prime Square and Valency Seven

Abstract: In this paper, we study symmetric graphs of valency seven and order 8p2, where p is an odd prime. It is proved that there are two graphs if the automorphism group of those graphs which is quasiprimitive on its vertices set, while it is no graphs exists in the case of the automorphism group is biquasiprimitive on the vertex set.

1. 引言

1) 假设X在 $V\Gamma$ 上是拟本原的，存在两个7度弧传递图 ${\mathcal{C}}_{72}^{1}$${\mathcal{C}}_{72}^{2}$，且 $Aut\left({\mathcal{C}}_{72}^{1}\right)\cong Aut\left({\mathcal{C}}_{72}^{2}\right)\cong PSL\left(2,8\right)×{ℤ}_{2}$

2) 假设X在 $V\Gamma$ 上是二部拟本原的，则图 $\Gamma$ 不存在。

2. 预备知识

$\Gamma$ 是阶为m的k度图，则 $|A\Gamma |=mk,|E\Gamma |=mk/2$。因此，当m为奇数时k为偶数。由此得出奇数阶对称图的度数为偶数。下面介绍两个具体的图。

a) $T=PSL\left(2,q\right)$，其中 $|\pi \left({q}^{2}-1\right)|=4$

b) $T=PSU\left(3,q\right)$，其中 $|\pi \left(\left({q}^{2}-1\right)\left({q}^{3}+1\right)\right)|=4$

c) $T=PSL\left(3,q\right)$，其中 $|\pi \left(\left({q}^{2}-1\right)\left({q}^{3}-1\right)\right)|=4$

d) $T={O}_{5}\left(q\right)$，其中 $|\pi \left({q}^{4}-1\right)|=4$

e) $T={S}_{z}\left({2}^{2m+1}\right)$，其中 $|\pi \left(\left({2}^{2m+1}-1\right)\left({2}^{4m+2}+1\right)\right)|=4$

f) $T=R\left({3}^{2m+1}\right)$，其中 $|\pi \left({3}^{4m+2}-1\right)|=3$$|\pi \left({3}^{4m+2}-{3}^{2m+1}+1\right)|=1$

g) $\begin{array}{l}T={A}_{11},{A}_{12},{M}_{22},{J}_{3},AS,{H}_{e},McL,PSL\left(4,4\right),PSL\left(4,5\right),PSL\left(4,7\right),PSL\left(5,2\right),PSL\left(5,3\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}PSL\left(6,2\right),{O}_{7}\left(3\right),PSp\left(6,3\right),PSp\left(8,2\right),PSU\left(4,4\right),PSU\left(4,5\right),PSU\left(4,7\right),PSU\left(4,9\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}PSU\left(5,3\right),PSU\left(6,2\right),{O}^{+}\left(8,3\right),{O}^{-}\left(8,2\right),{}^{3}D{}_{4}\left(3\right),{G}_{2}\left(4\right),{G}_{2}\left(5\right),{G}_{2}\left(7\right)或{G}_{2}\left(9\right)\end{array}$

7度连通对称图的点稳定子群在 [20] [定理1.1]和 [21] [定理3.4]中被独立确定出来，其中 ${F}_{n}$ 表示阶为n的Frobenius群，n为正整数。

1) 若 ${X}_{\alpha }$ 是可解的，则 $|{X}_{\alpha }||{2}^{2}\cdot {3}^{2}\cdot 7$。进一步， $\left(s,{X}_{\alpha }\right)$ 如下表1

Table 1. Solvable cases of stable subgroups of 7-degree graph points

2) 若 ${X}_{\alpha }$ 是不可解的，则 $|{X}_{\alpha }||{2}^{24}\cdot {3}^{4}\cdot {5}^{2}\cdot 7$。进一步， $\left(s,{X}_{\alpha }\right)$ 如下表2

Table 2. Unsolvable cases of stable subgroups of 7-degree graph points

1) N在 $V\Gamma$ 上半正则， $\frac{X}{N}\le Aut{\Gamma }_{N}$, ${\Gamma }_{N}$ 是X/N-弧传递的，且 $\Gamma$${\Gamma }_{N}$ 的正规N-覆盖；

2) $\Gamma$$\left(X,s\right)$ -弧传递的当且仅当 ${\Gamma }_{N}$$\left(X/N,s\right)$ -弧传递的，其中 $1\le s\le 5$$s=7$

3) ${X}_{\alpha }\cong {\left(X/N\right)}_{\delta }$，其中 $\alpha \in V\Gamma ,\delta \in V{\Gamma }_{N}$

3. 相关引理

1) 若 $|\pi \left(T\right)|=3$，则满足条件的 $\left(T,|T|,{p}^{2}\right)$ 如下表3

Table 3. Cases with 3 prime factors in a single group

2) 若 $|\pi \left(T\right)|=4$，则群T不存在。

$|\pi \left(T\right)|=4$, $|T||{2}^{27}\cdot {3}^{2}\cdot 7{p}^{2}$，且 $p>7$$p=5$ 从而有

${3}^{3}\nmid |T|$, ${7}^{2}\nmid |T|$, ${p}^{3}\nmid |T|$. (1)

$|PSL\left(2,r\right)|=\frac{r\left(r-1\right)\left(r+1\right)}{2}$

1) 若 $|\pi \left(T\right)|=3$，群T不存在。

2) 若 $|\pi \left(T\right)|=4$，满足条件的 $\left(T,|T|,{p}^{2}\right)$表4

3) 若 $|\pi \left(T\right)|=5$，群T不存在。

Table 4. The situation of 4 prime factors in a single group T

${2}^{12}\nmid |T|$, ${3}^{7}\nmid |T|$, ${5}^{5}\nmid |T|$, ${7}^{4}\nmid |T|$, ${p}^{2}||T|$.(2)

${2}^{12}\nmid |T|$, ${3}^{5}\nmid |T|$, ${5}^{3}\nmid |T|$, ${7}^{2}\nmid |T|$, ${p}^{2}||T|$ (3)

$|T|=\frac{q\left(q-1\right)\left(q+1\right)}{2}$

$\frac{q\left(q-1\right)\left(q+1\right)}{2}|{2}^{11}\cdot {3}^{4}\cdot {5}^{2}\cdot 7{p}^{2}$

$\frac{\left(q-1\right)}{2}\cdot \frac{\left(q+1\right)}{2}|{2}^{10}\cdot {3}^{4}\cdot {5}^{2}\cdot 7$.

$|T|=\frac{1}{\left(3,q+1\right)}{q}^{3}\left(q-1\right){\left(q-1\right)}^{2}\left({q}^{2}-q+1\right)$

$|T|=\frac{1}{\left(3,q-1\right)}{q}^{3}{\left(q-1\right)}^{2}\left(q+1\right)\left({q}^{2}+q+1\right)$

$|T|=\frac{1}{2}{q}^{4}\left({q}^{4}-1\right)\left({q}^{3}-1\right)\left({q}^{2}-1\right)$

4. 定理1.1的证明

$\Gamma$ 是连通的阶为8p2的7度X-弧传递图，其中 $X\le Aut\Gamma$，p是奇素数。设N是X的极小正规子群，则 $N={T}^{d}$，其中T为单群且 $d\ge 1$。设 $\alpha \in V\Gamma$。我们先证明下面的引理。

1) ${X}^{+}$${V}_{i}$ 上是拟本原的。

2) ${X}^{+}$ 有两个正规子群 ${U}_{1}$${U}_{2}$，使得 ${U}_{1}\cong {U}_{2}$$V\Gamma$ 上半正则。进一步可得 ${U}_{1}×{U}_{2}$${V}_{i}$ 上是正则的。

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