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

*g*_{1} 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 :/