Cubic Graphical Regular Representations of $\mathrm{PSU}_3(q)$

Cubic graphical regular representations of $PSL_2(q)$

We study cubic graphical regular representations of the finite simple groups $PSL_2(q)$. It is shown that such graphical regular representations exist if and only if $q\neq7$, and the generating set must consist of three involutions.

Cubic graphical regular representations of some classical simple groups

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 $1 \le i < ef$ t…

通过图小消去构造 $FP$ 型群

Groups of type $FP$ via graphical small cancellation

本文利用 Gromov 图小消去理论，而非 Morse 理论，构造了不可数多个 $FP$ 型群，提供了新的构造方法。