# of automorphisms of a cyclic group

[R.P.F.]

Homework Statement

Find all the automorphisms of a cyclic group of order 10.

Homework Equations

The Attempt at a Solution

I think it might be useful if I could first figure out how many automorphisms are there. There are 4 elements of order 5, 4 elements of order 10, 1 element of order 1 and 1 element of order 4. So is the # of automorphisms 4!4! ?

 

[Office_Shredder]

Try thinking about the automorphisms as mapping the generator of the group to another element

 

[R.P.F.]

Try thinking about the automorphisms as mapping the generator of the group to another element

Yeah I just realized that once the generator is picked, then the whole group is well defined. So there should be 4 automorphisms?