Critique of an Abstract Algebra proof

Question

I took Analysis last semester and did ok in it. I am now taking abstract algebra, and unfortunately didn't practice any math over the summer. Ever since I started taking upper level courses, doing proofs/showing something is true has been the hardest for me. Here is a proof from my abstract algebra textbook that I did, I am interested in any critique on it and what I could do to make it better.

1.3.2. Consider the symmetries of the square card.

(b) Observe that $r^3$ is the same symmetry as the rotation by $\frac{\pi}{2}$ 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 $r^{-k}=\left(r^{-1}\right)^k$ for any positive integer $k$. Show that $r^{-k}=r^{3 k}=r^m$, where $m$ is the unique element of $\{0,1,2,3\}$ such that $m+k$ is divisible by 4 .

Here is my "proof" of it.

Proof. Suppose $r^{3} = r^{-1}$ and define $r^{-1}$ to be the inverse. Define $r^{-k} = (r^{-1})^k$ for all $k \in \mathbb{Z}^+$. We want to show that $r^{-k}=r^{3 k}=r^m$. Whenever $k = 4z$, such that $z \in \mathbb{Z}+$, $r^{-k} = r^{3k} = e$, where $e$ is the non motion identity. Next, note that upon inspection $r^{-1} = r^3$ as defined, $r^{-2} = r^2$, and $r^{-3} = r^1$. Therefore any $k \neq 4z$ will have the form $k = (1+4z) \mod{4}$, $k = (2+4z) \mod{4}$, $k = (3+4z) \mod{4}$. Therefore each case holds for all $k$, and $r^{-k} = r^{3k}$.

Next, I choose to show that $r^{-k} = r^m$. This is equivalent to $(r^{-1})^k = r^m$. Whenver $k=4z$, then $m$ must equal $0$, and $(r^{-1})^k = r^m = 0$. For any other case we know that $m$ must be divisible by $4$, therefore $m+k=4$, or $m=4z-k$, in order to compensate for any z full rotations the circle makes. Substituting $m=4z-k$, we obtain:

$(r^{-1})^k = r^{4z-k}$

which is true. Therefore $r^{3k} = r^{m}$ and $r^{-k} = r^{3k} = r^{m}$.

This is a pretty simple proof. Am I going way overboard here with this? I'm not sure if my last substitution necessarily verifies the desired equation. I would greatly appreciate any thoughts on this.

Answer 1

It has several problems:

• You begin with “Suppose $r^3=r^{-1}$”. There is nothing to suppose here. You were told that $r^3=r^{-1}$.
• Then you wrote “and define $r^{-1}$ to be the inverse”. You had already mentioned $r^{-1}$ and it makes no sense to mention something first and only later to define it. Anyway, there is no need to define $r^{-1}$; its meaning is standard in the context of group theory.
• Then you write “Define $r^{−k}=(r^{−1})^k$ for all $k\in\Bbb Z_+$”. Again, this is part of the statement of the problem.

Then you complicate things a lot in order to prove that $r^{-k}=r^{3k}$. That's easy:$$r^{-1}=r^3\iff\left(r^{-1}\right)^k=\left(r^3\right)^k\iff r^{-k}=r^{3k}.$$

Finally, in order to prove that $r^{-k}=r^m$ where $m$ is the unique element of $\{0,1,2,3\}$ such that $m+k$ is divisible by $4$, note that $r^{m+k}=e$ and that\begin{align}r^{m+k}=e&\iff r^mr^k=e\\&\iff r^m=r^{-k}.\end{align}