I need help with this question:
(each variable $r$ represents a rotation of the square about the axis through its centroid at $90^{\circ}$ intervals. $e$ represents nonmotion.
This question is taken from the book Algebra: Abstract and Concrete found here. The question comes from Chapter $1$, Section $1.3$, Exercise $1.3.2$.
Here goes:
Consider the symmetries of the square card.
(a) Show that any positive power of $r$ must be one of $\{e,r,r^2, r^3\}$. First work out some examples, say through $r^{10}$. Show that for any natural number $k$, $r^k = r^m$, where $m$ is the nonnegative remainder after division of $k$ by $4$.
(b) Observe that $r^3$ is the same symmetry as the rotation by $90^{\circ}$ about the axis through the centroid of the faces of the square, in the clockwise sense, looking from the top of the square; that is, $r^3$ is the opposite motion to $r$, so $r^3 = r^{-1}$.
Define $r^{-k} = (r^{-1})^k$ for any positive integer $k$.
Show that $r^{-k} = r^{3k} = r^m$, where $m$ is the unique element of $\{0, 1,2,3\}$ such that $m+k$ is divisible by $4$.
End of problem.
I think I understand part a quite well, but I really need help with part b. Someone please help, because I'm stumped.
Thanks
First notice that by definition, $$ r^{-k} = (r^{-1})^k$$ and notice that you are given at the start of part b that $r^{-1} = r^3$. Inserting this identity, you have that $$r^{-k} = (r^3)^k$$
From here, look at the possible values for k (ie go case by case. There's only four cases). What is the remainder of 3k for each of these values? What happens when you add k and the remainder of 3k?