Interpreting a group homomorphism $f: \mathbb{Z}_{12} \to \mathbb{Z}_{3}$ visually

Question

I am having a hard time studying and I am a visual learner. How could I visually imagine a (group) homomorphism $$\mathbb{Z}_{12} \to \mathbb{Z}_3?$$

Also, if the question states that the map $f$ is a group homomorphism such that $f(1)=2$, how could I find the kernel $K$ of $f$?

Answer 1

I find it useful to imagine the group $\mathbb{Z}_n$ a group of rotations (of, e.g., the plane) by multiples of $\frac{1}{n}$ revolution, that is, by multiples of $\frac{2\pi}{n}$ radians, so that $[k] \in \mathbb{Z}_n$ corresponds to a (say, anticlockwise) rotation by $\frac{2 \pi k}{n}$ radians.

Since $\mathbb{Z}_n$ is cyclic, any group homomorphism $f: \mathbb{Z}_n \to H$ is determined by $f(1)$. This leaves three candidate maps, namely - the map defined by $f([1]) = [0]$, which is just the zero homomorphism $f([n]) := 0$, - the map defined by $f([1]) = [1]$, - the map defined by $f([1]) = [2]$.

Per the above mnemonic, we can regard $[1] \in \mathbb{Z}_{12}$ as an anticlockwise rotation by $\frac{1}{12}$ of a revolution, or $\frac{\pi}{6}$ radians, and, e.g., $[2] \in \mathbb{Z}_{3}$ as an anticlockwise rotation by $\frac{2}{3}$ of a revolution, or $\frac{4\pi}{3}$ radians. So, using that $f$ is a group homomorphism, so that $f([n]) = n \cdot f([1])$ we can think of $\phi$ as the map that takes a given rotation in $\mathbb{Z}_{12}$ and applies it $\frac{\frac{2}{3} \text{ rev}}{\frac{1}{12} \text{ rev}} = 8$ times, which by construction is always a rotation in $\mathbb{Z}_3$ so imagined.

(As a word of warning, this is a convenient way to think about finite cyclic groups, but if one wants to think about $\mathbb{Z}_3$ sitting inside $\mathbb{Z}_{12}$ the way we did here, one must specify that one is using this mnemonic; indeed, there is more than one way to put $\mathbb{Z}_3$ into $\mathbb{Z}_{12}$ in a way that respects group multiplication.)

Thinking of a group as we did here as a set of (linear) transformations of some vector space, by the way, is a (the) central idea of representation theory, which has proved to be an immensely powerful too for understanding groups.