Pentavalent Symmetric Graphs Of Order 100p And On Sporadic Simple Group J₁ Covered By Its Abelian Subgroups

An s-arc in a graph ? is an ordered(s + 1)-tuple(v0,v1,...,vs),such that vi-1 is adjacent to vi and vi-1 ? vi+1 for 1 ? i ? s.A graph ? is said to be(G,s)-arc-transitive if G acts transitively on the set of s-arcs,where G ? Aut(?).A graph ? is said to be s-arc-transitive if G = Aut(?).In this paper,1-arc-transitive graph,which is also called symmetric graph,is studied.More specifically,we mainly research the connected pentavalent symmetric graph of order 100p with it's automorphism group having no solvable minimal normal subgroups.As a re-sult,we show that there are no such graphs.Namely,if the connected pentavalent symmetric graph of order 100p exists,then it's automorphism group must have a solvable minimal normal subgroup.The abelian subgroup cover number of a group G is defined to be the least number n of proper abelian subgroups of G whose union is equal to G.In this paper,we prove that the abelian subgroup cover number of J1 is 33650.