Prove that [tex]Aut(Z_2\oplus Z_4) \cong D_8[/tex]
The Attempt at a Solution
I wrote out all of the elements of [tex]Z_2\oplus Z_4[/tex]. There are 8 of them, of course. Then I need to find the automorphisms of it. It looks to me like they would be the same as Aut(D8),which I have. I have proved (I think) that [tex]Aut(D_8)\cong D_8[/tex]. I'm just unsure HOW to go about this one.
Could I show with a cayley table that [tex]Z_2\oplus Z_4[/tex] is nonabelian and it's clearly of order 8, so then the isomorphism holds?
I just get stuck on these types of problems. the idea of isomorphisms is still pretty fresh to me and so are the groups themselves.
Any help will be appreciated.