Determining the order of elements in a group

Question

Given a group table, determine the order of $x_5, x_3, x_5x_3, (x_5)^2(x_3), x_5^{-1}x_3$.

So the order of an element is the least positive integer n such that $a^n = e$. So $x_5 = 2$ and $x_3 = 2$ by inspection of the table. I'm confused at how you would calculate the other ones though.

Answer 1

• Using the table, the element $x_5x_3=x_5 \cdot x_3$ corresponds to $x_6$, thus $$x_5x_3=x_6 \implies |x_5x_3|=|x_6|$$ that is, the order of $x_6$ is that of $x_5x_3$ because they're the same element.

• The element $(x_5)^2(x_3)=x_5\cdot x_5\cdot x_3 = (x_5 \cdot x_5) \cdot x_3 = e \cdot x_3 = x_3$. So $|(x_5)^2(x_3)|=|x_3|$. What's $|x_3|$ then?

• For the final one, note $x_5^{-1}$ is what gets multiplied with $x_5$ resulting in $e$. That is, trace down the row/column of $x_5$ and look for an $e$. What was multiplied with $x_5$ to result in that $e$? That's your $x_5^{-1}$. Also, the reason you can use either the row/column of $x_5$ is because $x_5 x_5^{-1} = x_5^{-1} x_5 = e$, think about it! Once you found $x_5^{-1}$, find $x_5^{-1}x_3$. You know where to go from there.

Answer 2

One has $x_5x_3 = x_6$, then $ord(x_5x_3) = ord(x_6) = 2$.

$x_5^2x_3 = x_3$, then $ord(x_5^2x_3) = ord(x_3) = 2$.

$x_5^{-1}x_3 = x_5x_3$, then $ord(x_5^{-1}x_3) = ord(x_5x_3) = 2$.