"The group S6 acts on the group Z6 via σ([a]) = [σ(a)], for σ ∈ S6 and a∈{1,...,6}.
A permutation that is also an isomorphism is called an automorphism. The set G of automorphisms of Z6 is a group. Use the orbit-stabiliser theorem to find its order."
I am considering the element [1], and I have found that the size of its orbit is 2 (as isomorphisms preserve orders of elements, [1] may only be mapped to itself or [5]). However, I have no idea how to go about finding the size of the stabiliser of [1]. Some help would be greatly appreciated!
Hint: Recall that for any homomorphism $\pi$, the value of $\pi(g)$ determines the value of $\pi$ on all of $\langle g \rangle$. What does that mean in your situation?