The automorphism group of a graph &Gamma is the collection of adjacency-preserving permutations of the vertices, denoted by Aut &Gamma. Any finite abstract group can be realized as the automorphism group of a graph. Here our purpose is to minimize graph invariants under certain symmetric restrictions. We construct vertex-minimal planar graphs with abelian automorphism groups. Further, we classify vertex-minimal graphs whose automorphism group is isomorphic to a modular p-group, where p is an odd prime. The fixing number of a graph &Gamma is defined to be the smallest number of vertices in &Gamma that, when fixed, makes Aut &Gamma trivial. For a finite group G, the fixing set is defined as the set of all fixing numbers of graphs having automorphism groups isomorphic to G. We discuss the fixing set of modular p-group.