## 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 ofare given below

Element | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Order | 0 | 10 | 5 | 10 | 5 | 2 | 5 | 10 | 5 | 10 |

Ifis an automorphism ofthen 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 asis decided,is fixed sinceand similarly for all the other elements.

Call e the element

Call a the element

Call b the element

Call c the element

Thensincand since

The group of automorphisms ofis cyclic, with four elements, and is isomorphic to