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

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}

Groups having complete bipartite divisor graphs for their conjugacy class sizes

Given a finite group G, the bipartite divisor graph for its conjugacy class sizes is the bipartite graph with bipartition consisting of the set of conjugacy class sizes of G-Z (where Z denotes the centre of G) and the set of prime numbers that divide these conjugacy class sizes, and with {p,n} being an edge if gcd(p,n)\neq 1. In this paper we construct infinitely many groups whose bipartite divisor graph for their conjugacy class sizes is the complete bipartite graph K_{2,5}, giving a solution to a question of Taeri.Comment: 5 page