882 research outputs found
On graph-restrictive permutation groups
Let be a connected -vertex-transitive graph, let be a vertex
of and let be the permutation group induced by the
action of the vertex-stabiliser on the neighbourhood . Then
is said to be \emph{locally-}. A transitive permutation group
is \emph{graph-restrictive} if there exists a constant such that,
for every locally- pair and an arc of , the
inequality holds.
Using this terminology, the Weiss Conjecture says that primitive groups are
graph-restrictive. We propose a very strong generalisation of this conjecture:
a group is graph-restrictive if and only if it is semiprimitive. (A transitive
permutation group is said to be \emph{semiprimitive} if each of its normal
subgroups is either transitive or semiregular.) Our main result is a proof of
one of the two implications of this conjecture, namely that graph-restrictive
groups are semiprimitive. We also collect the known results and prove some new
ones regarding the other implication
On the order of arc-stabilisers in arc-transitive graphs with prescribed local group
Let be a connected -arc-transitive graph, let be an arc of
and let be the permutation group induced by the action of the
vertex-stabiliser on the neighbourhood . We study the problem
of bounding in terms of and the order of .Comment: 17 pages, 1 tabl
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