## The Group of Automorphisms of the Addition Group Z10

The set of automorphisms of a group is itself a group. Since automorphisms preserve order and inverses, we can analyse the orders of each element to decide on possible automorphisms.

The orders of each element of are given below

 Element 0 1 2 3 4 5 6 7 8 9 Order 0 10 5 10 5 2 5 10 5 10

If is an automorphism of then we can sort the elements into sets by the elements they can be mapped to by an automorphism. Each element in the same set has the same order.

{0}, {1, 3, 7, 9}, {2, 4, 6, 8} and {5}.

If 1 is mapped to 3 then 1+1=2 is mapped to 3+3=6, 1+1+1=3 is mapped to 3+3+3=9, 1+1+1+1=12 is mapped to 3+3+3+3=12 ≡ 2 (mod 10). Continuing in this way we obtain the mapping Similarly, if 1 is mapped to 7, If 1 is mapped to 9, As soon as is decided, is fixed since and similarly for all the other elements.

Call e the element Call a the element Call b the element Call c the element Then sinc and since The group of automorphisms of is cyclic, with four elements, and is isomorphic to  