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## Homework Statement

Find, with justification, all the elements of ##\operatorname{Aut} (\mathbb{Z}/4\mathbb{Z})##. Which ones are in ##\operatorname{Inn} (\mathbb{Z} / 4\mathbb{Z})##?

## Homework Equations

## The Attempt at a Solution

First we note that ##\operatorname{id}_{\mathbb{Z} / 4\mathbb{Z}} \in \operatorname{Aut} (\mathbb{Z}/4\mathbb{Z})## and that it is the only element in ##\operatorname{Inn} (\mathbb{Z} / 4\mathbb{Z})##. This is because any inner automorphism will have the form ##c_g (x) = g+ x - g##. But since ##\mathbb{Z} / 4\mathbb{Z}## is abelian the g's cancel out and we're left with the identity.

Now, we claim that there is only one other automorphism. This comes from the the fact that ##|0| = 1,~|1|=4,~|2| = 2,~|3|=4##. Since order must be preserved by a homomorphism, ##3## must either mapped to ##1## or itself. Likewise, ##1## must be mapped to itself or ##3##. ##0## must be mapped to itself (as it is a property of homomorphisms). ##2## must also be mapped to itself because it is the only element of order 2. This leaves us with the identity map and the map where ##f(0)=0,~f(1)=3,~f(2)=2,~f(3)=1## as possibilities for automorphisms. This map is certainly bijective, so we must only check that it is a homomorphism. Since ##\mathbb{Z} / 4\mathbb{Z}## is abelian we only have to check three sums: ##f(1+2) = f(3) = 1~|~1=3+2=f(1)+f(2)##

##f(1+3) = f(0) = 0~|~0=4=3+1=f(1)+f(3)##

##f(2+3) = f(5)= f(1) = 3~|~3=2+1=f(2)+f(3)##. Hence, the map ##f## is an automorphism.