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### Problem 5

An **automorphism of a group** is an isomorphism of a group with itself. Find the number of automorphisms of the following groups. (Hint: Make use of number 2. What must the image of a generator be under an isomorphism?)

(a) $\mathbb{Z}_{6}$

(b) (+) $\mathbb{Z}_{20}$

**Solution**

(a) The automorphisms of a group will be given by the generators of the group, as 1 must map to another generator and an isomorphism is completely defined by what it does to 1. The generators for $\mathbb{Z}_{6}$ are $\langle 1 \rangle$ and $\langle 5 \rangle$. Thus, the number of automorphisms for $\mathbb{Z}_{6}$ is $2$.

(b) The automorphisms of a group such as $\mathbb{Z}_{20}$ will be just given by the generators of the group as 1 must map to another generator and an isomorphism is completely defined by what it does to 1.. The generators for $\mathbb{Z}_{20}$ are $\langle 1 \rangle, \langle 3 \rangle, \langle 7 \rangle, \langle 9 \rangle, \langle 11 \rangle, \langle 13 \rangle, \langle 17 \rangle$ and $\langle 19 \rangle$. Then the number of automorphisms for $\mathbb{Z}_{20}$ is 8.