Prove that Aut(V) is isomorphic to and that is isomorphic to . V is the four group.
Also, prove that Aut(Z) is isomorphic to . Z is the set of integers and is the integers mod 2.
I appreciate any help.
1) Any automorphism on V will preserve order of each elements. Therefore it can permute i,j,k. There are no restrictions on the permutation. Hence Aut(V)= .
2) Any isomorphism from the integers will have the form (*)
Let
Then for all positive integers (induction using (*)
And for negative integers
For this to be a bijection and we're done