Task: Find the number of automorphisms in the group [MATH]\mathbb{Z}_{5}[/MATH].
My approach: First I notice that there are 5! possible bijective maps for a set of 5 elements. An automorphism maps the identity element to itself, so the number narrows down to 4!. Next I tried to argue with generators/order of the elements which are preserved for an automorphism to further narrow down the number of possibilieties. 1, 2, 3 and 4 are all generators of order 5. I don't see how to proceed from here.
My approach: First I notice that there are 5! possible bijective maps for a set of 5 elements. An automorphism maps the identity element to itself, so the number narrows down to 4!. Next I tried to argue with generators/order of the elements which are preserved for an automorphism to further narrow down the number of possibilieties. 1, 2, 3 and 4 are all generators of order 5. I don't see how to proceed from here.