The task is to describe all isometries of the square according to the following steps:
(a) What is a fixed point of all isometries of the square? What types of isometries does the
square have in terms of fixed points?
(b) Show that if three vertices of a nondegenerate triangle are fixed by an isometry then the
isometry is trivial.
(c) How many different isometries does the square have?
(d) Show that those isometries that move A to A and those that move A to B (respectively,
to C, respectively, to D) are in bijection. How can this be used to obtain part (c) in an
alternative way?
My solution:
(a) The center of the square is fixed in all isometries. In terms of fixed points the square has the following isometries: identity isometry (all points are fixed), reflection about main diagonal line, second diagonal line, horizontal line passing through the center, vertical line passing through the center ( in all reflections, points lying on the reflection lines remain unchanged).
(b) Let $A, B, C$ be vertices of the nondegenerate triangle. Let $ \phi $ be the isometry. Then $ \phi(A) = A, \phi(B)=B, \phi(C)=C $ by the assumption. Trivial isometry is identity ( $Id$), identity maps $A$ to $A$, $B$ to $B$ and $C$ to $C$. Let $ \psi$ be isomorphic transformation that sends $A$ to some vertex $A_\psi $, $B$ to $B_\psi$, $C$ to $C_\psi$, where $A_\psi, B_\psi, C_\psi $ are pairwise non-equal to each other. Then, $\phi(\psi) = \psi = Id \psi$ $ \Rightarrow Id = \phi$.
(с) The square has 8 different isometries. Namely, $R_0, R_{\pi/2}, R_\pi, R_{3\pi/2}, D, D', H, V $, where the last four are reflections about main diagonal, second diagonal, horizontal line through center, vertical line through center, respectively, and first four are rotation by angles $0, \pi/2, \pi, 3\pi/2$, respectively. It is in fact, dihedral group of order 4 ($D_4$)
(d) These isometries are $R_0, R_{\pi/2}, R_\pi, R_{3\pi/2}$. Obviously, the relation is injective. To show that they are in surjection, we need to show that there are four distinct isometries that move $A$ to $A$, $B$ to $A$, $C$ to $A$, $D$ to $A$, and they are the ones from this list $R_0, R_{\pi/2}, R_\pi, R_{3\pi/2}$. In fact $R_0$ sends $A$ to $A$, $R_{3\pi/2}$ sends $B$ to $A$, $R_\pi$ sends $C$ to $A$ and $R_\pi/2$ sends $D$ to $A$. We showed that they are in bijection. To obtain part (c) we take any different isometry, for example that, which sends $D$ to $B$, and $A$ to $A$. We multiply it by $R_0, R_{\pi/2}, R_\pi, R_{3\pi/2}$ and obtain other four isometries.