On the Minimum Order of Graphs with Given Group

Abstract

ForGa, finite group letα(G) denote the minimum number of vertices of the graphsXthe automorphism groupA(X) of which is isomorphic toG.G. Sabidussi proved [1], thatα(G)=0(nlogd) wheren=\G\anddis the minimum number of generators ofG.As0(log n) is the best possible upper bound ford,the result established in [1] implies thatα(G)=0(nlog logn).