consider the group G of all symmetries of an equilateral triangle. find the order of every element of this group

Question

consider the group G of all symmetries of an equilateral triangle. find the order of every element of this group enter image description here

Answer 1

Consider rotations ($r$) and reflections ($s$) $$\langle r,s \mid r^3=s^2=e\,,\,srs=r^{-1}\rangle$$

Which has order $6$ (why?)

Notice that it makes sense to say that $r^3=e$ is a "cyclic group" of order $3$, since this is essentially just rotations by $2 \pi/3$, and three rotations get you back to where you started. On the other hand, $s^2$ also makes sense, since two reflections get you back to where you started.

The last relation can be re-written as $srsr=e$ or $rsrs=e$, but the other way of writing it is just more concise. You should figure out why this relation is necessary.

Answer 2

Perhaps the most helpful way to go about understanding this group is to draw it out and see how many variations of the triangle each symmetry can generate. Consider each symmetry individually first: the identity in which no movement occurs, a $120$ degree rotation, and a flip in which two corners are exchanged. It will be easy to see that the $120$ degree rotation alone will generate three elements and the flip will generate two elements. Now we have $6$ elements total (three rotations, two flips, $1$ identity). What happens when you combine these elements? Are any of the transformations the same? (Hint: as given in the previous answer for example, $r^3=s^2$. What does this mean geometrically?)