The fixing number of a graph \(\Gamma \) is the minimum number of vertices that, when fixed, remove all nontrivial automorphisms from the automorphism group of \(\Gamma \) . Gibbons and Laison extended the concept of fixing numbers of graphs to fixing sets of groups. The fixing set of a finite group G is the set of all fixing numbers of graphs whose automorphism groups are isomorphic to G. For \(n \in {\mathbb {Z}}\) with \(n \ge 2\) , Gibbons and Laison conjectured that the fixing set of the symmetric group on n symbols is \(\{1,2,\ldots , n-1\}\) . In this article, we will disprove this conjecture, which also answers an open question of theirs. Moreover, we will establish previously unknown elements of fixing sets of symmetric groups by using Johnson graphs. Fixing sets of other groups have also been considered; however, there are very few fixing sets of groups that have been completely established. We will begin this article by proving the fixing sets of both quasi-dihedral groups and quasi-abelian groups; these results then determine the fixing set of every group that is isomorphic to a member in one of the six infinite families of 2-groups that contain a cyclic subgroup of index 2.
