Two local conditions on the vertex stabiliser of arc-transitive graphs and their effect on the Sylow subgroups

In this paper we study $G$-arc-transitive graphs $\Delta$ where the permutation group $G_x^{\Delta(x)}$ induced by the stabiliser $G_x$ of the vertex $x$ on the neighbourhood $\Delta(x)$ satisfies the two conditions given in the introduction. We show that for such a $G$-arc-transitive graph $\Delta$, if $(x,y)$ is an arc of $\Delta$, then the subgroup $G_{x,y}^{[1]}$ of $G$ fixing pointwise $\Delta(x)$ and $\Delta(y)$ is a $p$-group for some prime $p$. Next we prove that every $G$-locally primitive (respectively quasiprimitive, semiprimitive) graph satisfies our two local hypotheses. Thus this provides a new Thompson-Wielandt-like theorem for a very large class of arc-transitive graphs. Furthermore, we give various families of $G$-arc-transitive graphs where our two local conditions do not apply and where $G_{x,y}^{[1]}$ has arbitrarily large composition factors

Abelian Carter subgroups in finite permutation groups

We show that a finite permutation group containing a regular abelian self-normalizing subgroup is soluble.Comment: 6 page

On the maximum order of nilpotent transitive permutation groups

Given two positive integers n and c, we determine an upper bound, as a function of n and c, for the maximum order of a finite nilpotent transitive group of degree n and nilpotency class at most c.Comment: 14 page

Maximal subgroups of finite groups avoiding the elements of a generating set

We give an elementary proof of the following remark: if G is a finite group and { g1, \u2026 , gd} is a generating set of G of smallest cardinality, then there exists a maximal subgroup M of G such that M 29 { g1, \u2026 , gd} = 05. This result leads us to investigate the freedom that one has in the choice of the maximal subgroup M of G. We obtain information in this direction in the case when G is soluble, describing for example the structure of G when there is a unique choice for M. When G is a primitive permutation group one can ask whether is it possible to choose in the role of M a point-stabilizer. We give a positive answer when G is a 3-generated primitive permutation group but we leave open the following question: does there exist a (soluble) primitive permutation group G= \u27e8 g1, \u2026 , gd\u27e9 with d(G) = dCloseSPigtSPi 3 and with c2 1 64 i 64 dsupp (gi) = 05? We obtain a weaker result in this direction: if G= \u27e8 g1, \u2026 , gd\u27e9 with d(G) = d, then supp (gi) 29 supp (gj) 60 05 for all i, j 08 { 1 , \u2026 , d}