Order of an element in direct product using cayley's diagram

Question

How can I find the order of element (1,1) of the group $C_4\times C_3$ visually in the diagram below :

Answer 1

Notice that you can get from $(0,0)$ to $(1,1)$ by following one red arrow and one blue arrow. Repeat this operation with the aid of the diagram until you get back to $(0,0)$, and count the number of steps. That will give you the order of $(1,1)$.

Answer 2

Multiplication by (1,1) is following a red arrow then a blue arrow.

Or you could think of it as moving across one column, down one row.

Start at (0,0), you're in the first column, every four moves you're back in the first column.

Every three moves you're back in the first row.

So if (1,1)^n is (0,0), then n has to be both a multiple of three, and a multiple of four.

The first number that's a multiple of three and a multiple of four is 12.

So the order of (1,1) is 12.

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More generally, if you have the same diagram, but m across and n down, instead of 4 across and 3 down, then the order of (1,1) will be the least common multiple of m and n.