论文标题
某些经典简单组的立方图形规则表示
Cubic graphical regular representations of some classical simple groups
论文作者
论文摘要
$ g $组的图形常规表示（GRR）是$ g $的开纱图，其完整的自动形态组等于$ g $的正确常规置换表示。在本文中，我们研究了$ \ mathrm {psl} _ {n}（q）$（$ n = 4、6、8 $），$ \ mathrm {psp} _ {n}（q）$（$ n = 6、8 $ n = 6、8 $），$ \ mathrm {pssp} $ n} $ n} $ n { 12 $）和$ \ mathrm {p}ω__{n}^{ - }（q）$（$ n = 8，10，12 $），其中$ q = 2^f $带有$ f \ ge 1 $。我们证明，对于每个组中的每个组，概率倾向于$ q \ rightarrow \ infty $，任何元素$ x $ ofd Prime订单的$ x $ sivise $ 2^{ef} -1 $，但不是$ 2^{i} -1 $ for $ 1 \ le I <ef $ cair y y y $ y $ y $ y $ y $ y $ y $ e = $ \ mathrm {p}ω__{n}^{+}（q）$和$ e = n $用于其他组。此外，对于足够大的$ q $，有元素$ x $满足这些条件，对于它们中的每个条件，都存在$ y $ $ y $，以便$ \ {x，x^{ - 1}，y \} $产生一个立方体的GRR。该结果与文献中的某些已知结果一起意味着，除了$ \ mathrm {psl} _2（q）$，$ \ mathrm {psl} _3（q）$，$ \ mathrm {psu} _3（q）_3（q）$和其他每个有限的$ y $ y $ y $ y $ $ \ {x，x^{ - 1}，y \} $产生一个grr，表明Spiga的猜想的修改版本是正确的。我们的结果和几个已知的结果共同证实了Fang和Xia的猜想，这些猜想断言，除了有限数量的每个有限的非亚伯利亚简单组有限的情况外，我们都有一个立方GRR。
A graphical regular representation (GRR) of a group $G$ is a Cayley graph of $G$ whose full automorphism group is equal to the right regular permutation representation of $G$. In this paper we study cubic GRRs of $\mathrm{PSL}_{n}(q)$ ($n=4, 6, 8$), $\mathrm{PSp}_{n}(q)$ ($n=6, 8$), $\mathrm{P}Ω_{n}^{+}(q)$ ($n=8, 10, 12$) and $\mathrm{P}Ω_{n}^{-}(q)$ ($n=8, 10, 12$), where $q = 2^f$ with $f \ge 1$. We prove that for each of these groups, with probability tending to $1$ as $q \rightarrow \infty$, any element $x$ of odd prime order dividing $2^{ef}-1$ but not $2^{i}-1$ for each $1 \le i < ef$ together with a random involution $y$ gives rise to a cubic GRR, where $e=n-2$ for $\mathrm{P}Ω_{n}^{+}(q)$ and $e=n$ for other groups. Moreover, for sufficiently large $q$, there are elements $x$ satisfying these conditions, and for each of them there exists an involution $y$ such that $\{x,x^{-1},y\}$ produces a cubic GRR. This result together with certain known results in the literature implies that except for $\mathrm{PSL}_2(q)$, $\mathrm{PSL}_3(q)$, $\mathrm{PSU}_3(q)$ and a finite number of other cases, every finite non-abelian simple group contains an element $x$ and an involution $y$ such that $\{x,x^{-1},y\}$ produces a GRR, showing that a modified version of a conjecture by Spiga is true. Our results and several known results together also confirm a conjecture by Fang and Xia which asserts that except for a finite number of cases every finite non-abelian simple group has a cubic GRR.
