Let G be a group and let g be some element of G. Given that is an isomorphism is such that , show that and are conjugate subgroups of G. Here, is the subgroup of the set of automorphisms of G that fix the element x. In other words, .
I haven't gotten anywhere with this after about 5 hours now. I feel like the key to the problem will be pulling in somehow- intuitively, g and g1 seem to me sort of like inverses of each other across the isomorphism. Since is a member of the group Aut(G), it has an inverse, say such that and I feel like somehow will be very involved in the conjugation of . I don't know. I've been staring at a whiteboard basically writing the definition of conjugation over and over and over and over without getting anywhere. I already have proved that is a subgroup of Aut(G). I'm stuck :/