How can group theory explain movement on a hexagonal tiling?

Question

(As a prelude, I have no formal math training other than high school. I am a beginner with group theory and have just recently begun picking it up and seeing its potential uses.)

Imagine an infinite game board whose spaces make a hexagonal tiling. Pieces on this board can move directly to adjacent spaces from their current space, or choose not to move at all (denoted as e). The six directions they can move could be called N, NE, SE, S, SW, and NW. This forms three distinct axes, where positive movement on axis a would correspond with N, positive movement on axis b would correspond with SE, and positive movement on axis c would correspond with SW.

Describing movement on this board meets the criteria of an abellian group:

• Closure: Any combination of movements will still land the piece on this infinite game board.
• Associativity: (a • b) • c will result in the same location as a • (b • c), although they will take different paths.
• Identity element: The zero movement (not moving at all) combined with any other movement results in just the other movement. (e • a = a, a • e = a)
• Inverse element: Moving one space towards N and then one space towards S results in zero net movement. (a • (-a) = e)
• Commutativity: Moving one space on the a axis and then one space on the b axis results in the same position as moving one space on the b axis and then one space on the a axis. (a • b = b • a)

The ways that movements combine is interesting, though. The three axes are not orthogonal to each other. For instance, a • b • c = e and a • b = -c for all axes.

I'm looking for a deeper understanding of the patterns that emerge from this. Does this group have a name? How can one use group theory to describe, for instance, an algorithm that can determine if two long lists of movements on the board result in the same net movement? (a • a • b • c • (-a) = e)

Answer 1

The group is isomorphic to $\mathbb{Z}^2$. Proof:

From JohnBarber's presentation in the comments above, the group is also isomorphic to the set of complex numbers generated by $a$ and $b$, defined as in his comment (since $a+b+c=0$). For $z \in G$, $z$ has a unique representation as $\lambda a + \mu b$, where $\lambda$, $\mu$ are integers (exercise: prove it's unique).

Then define a group homomorphism $\phi$ from $G$ to $\mathbb{Z}^2$ by $\phi(a) = (1, 0)$ and $\phi(b) = (0, 1)$. The homomorphism is well defined since the representation of $z$ was unique and $\phi$ is defined on the generators.

Also, the homomorphism is injective (check the kernel), and surjective (look at $ma+nb$), so it is an isomorphism.

Notice that $\phi(c) = (-1, -1)$. So, for some sequence of moves, convert $a$ to $(1,0)$, $b$ to $(0,1)$ and $c$ to $(-1, -1)$ and add the points.