Simple graphs containing induced subgraphs whose automorphism groups are isomorphic to subgroups of a given finite group Original Research Article

Given a finite group A, Frucht has given a method to construct a simple graph G whose automorphism group Aut(G) is isomorphic to to A [1]. In this paper, we show that by selecting a suitable generating set for A, the Frucht graph G obtained by the Frucht construction satisfies the following property: to each subgroup H of A, there is an induced subgraph GH of G such that Aut(GH)≃H and the association H →H is inclusion preserving. We also construct a graph G such that Aut(G)={e} and to each subgroup H of~A, there is an induced subgraph GH of G such that Aut(GH)≃H and the association H→GH is inclusion reversing.