Can someone explain this solution to me?
The goal is to find an isomorphism from $ [\mathbb{Z}_6, \oplus_6]$ to $[\mathbb{Z}_7\setminus\{0\}, \otimes_7]$, where ${}\setminus\{0\}$ means "excluding 0".
There are 2 isomorphisms:
$f(0) = 1, \quad f(1)=3, \quad f(2)=2, \quad f(3)=6, \quad f(4)=4, \quad f(5)=5$
and
$f(0) = 1, \quad f(1)=5, \quad f(2)=4, \quad f(3)=6, \quad f(4)=2, \quad f(5)=3$
I don't understand part of the solution...
I know the generators for $\mathbb{Z}_6$ are {$1$, $5$}. Similarly, the generators for $\mathbb{Z}_7\backslash\{0\}$ are {$3$, $5$}.
To find the isomorphism, we're supposed to map corresponding powers of generators like so: $f(a^k)=b^k$, where $a$ is the generator for $\mathbb{Z}_6$ and $b$ is the generator for $\mathbb{Z}_7 \backslash\{0\}$.
For the first isomorphism, we use $a=1$, $b=3$. For the second isomorphism, we use $a=1$, $b=5$. How am I supposed to know which $a$ and which $b$ to use? For example, how come there isn't an isomorphism that uses $a=5$, $b=3$? I know the $1$, $3$, and $5$ come from finding the generators, but I don't understand why the 2 isomorphisms use $a=1$, $b=3$ and $a=1$, $b=5$ in particular.
Hint: If $1$ is mapped to $5,$ then what is $5$ mapped to? If $1$ is mapped to $3,$ then what is $5$ mapped to?
The kicker, here, is to pick a consistent generator, and decide what it gets mapped to. It doesn't actually matter which consistent generator gets picked, so long as we remain consistent. Based on the answer to the above questions, can you see why consistency is important?