Earlier, to confirm that one of the possibilities for the structure of vertex stabilizers of graphs with projective suborbits is realizable, we announced the existence of a connected graph
Γ
...admitting a group of automorphisms
𝐺
which is isomorphic to Aut
𝐹
subscript
𝑖
22
and has the following properties. First, the group
𝐺
acts transitively on the set of vertices of
Γ
, but intransitively on the set of
3
-arcs of
Γ
. Second, the stabilizer in
𝐺
of a vertex of
Γ
induces on the neighborhood of this vertex a group
𝑃
𝑆
subscript
𝐿
3
3
in its natural doubly transitive action. Third, the pointwise stabilizer in
𝐺
of a ball of radius 2 in
Γ
is nontrivial. In this paper, we construct such a graph
Γ
with
𝐺
Aut
Γ
.
On the Weiss Conjecture. I Trofimov, V. I.
Proceedings of the Steklov Institute of Mathematics,
12/2022, Volume:
319, Issue:
Suppl 1
Journal Article, Conference Proceeding
Peer reviewed
Let
Γ
be a connected finite graph, and let
𝐺
be a vertex-transitive group of automorphisms of
Γ
such that the stabilizer
subscript
𝐺
𝑥
in
𝐺
of a vertex
𝑥
of
Γ
induces on the neighborhood
Γ
𝑥
...of
𝑥
a primitive permutation group
superscript
subscript
𝐺
𝑥
Γ
𝑥
. The Weiss conjecture says that, under these assumptions, the order of
subscript
𝐺
𝑥
is bounded from above by a number depending only on the order
Γ
𝑥
of
Γ
. In a research whose first part is the present paper, we show that some general results of the theory of finite groups can be used to provide a largely uniform analysis for a number of cases of the Weiss conjecture (including some cases that were not considered before). Although this first part is introductory, it makes possible to use certain previous results to confirm the Weiss conjecture for all primitive groups
superscript
subscript
𝐺
𝑥
Γ
𝑥
different from groups of AS type and from groups of PA type (constructed on the basis of groups of AS type).
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