We determine the full automorphism group of every connected Cayley graph on an almost simple group with a normal connection set. We also characterize exactly when the full automorphism group is generated by right translations, group automorphisms preserving the connection set, and inversion. Our results substantially generalize several results in the literature, including the known determinations of the automorphism groups of derangement graphs, complete transposition graphs and complete alternating group graphs. As a further application, we establish criteria for a finite nonabelian simple group to admit a graphical doubly regular representation and determine exactly which alternating groups do so, correcting claims in the literature.
We show that if $\varphi$ is a colour-permuting automorphism of a connected, finite Cayley graph, and the order of the Cayley graph is either odd or square-free, then $\varphi$ is the composition of a group automorphism and a colour-preserving graph automorphism. Some analogous results are also established for isomorph...
We prove that for every ergodic transformation $T$ on an infinite standard probability space both the automorphism group (i.e. centralizer) $\mathrm{Aut}(T)$ and its reversing automorphism group are realizable as symmetry groups of graphings. This is an analogue of Sabidussi's realization of arbitrary graph-automorphis...
Let $\Ga$ be a connected prime-valent $X$-arc-transitive Cayley graph of a finite characteristically simple group $G\cong T^k$, where $k\geqslant2$. We obtain a precise structural characterization of such graphs and their arc-transitive automorphism groups. In the cubic case, every connected symmetric Cayley graph of $...
In this paper, we consider the groups of linear operators that preserve idempotent matrices over the binary Boolean algebra. We prove that this set forms an automorphism group under composition and determine its exact algebraic structure as a direct product Sn×C2 for n≥2. Furthermore, we distinguish between elementary...
We study graph products of groups over defining graphs of arbitrary cardinality from two closely related viewpoints. First, we characterize acylindrical hyperbolicity. If the defining graph is irreducible, has at least two vertices, and has a finite star base, then every parabolically full subgroup is either virtually...
The group of polynomial automorphisms of the affine n-space is an interesting large group. A slightly simpler group is its subgroup of tame automorphisms. Natural problems about these groups include the existence of normal subgroups, the classification of finite subgroups, the Tits alternative, and the possible dynamic...
St'ephane Lamy· 0 citations
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