$\newcommand{\nfrac}[2]{\frac{\displaystyle{#1}}{\displaystyle{#2}}}$
Problems
2
| R_0 | R_120 | R_240 | D | D' | D'' | |
|---|---|---|---|---|---|---|
| R_0 | R_0 | R_120 | R_240 | D | D' | D'' |
| R_120 | R_120 | R_240 | R_0 | D'' | D | D' |
| R_240 | R_240 | R_0 | R_120 | D' | D'' | D |
| D | D | D' | D'' | R_0 | R_120 | R_240 |
| D' | D' | D'' | D | R_240 | R_0 | R_120 |
| D'' | D'' | R_0 | D' | R_120 | R_240 | R_0 |
Two pictures.
Not abelian.
3
a. $V$.
b. $R_{270}$.
c. $R_0$.
d. $R_0$, $R_{180}$, $H$, $V$, $D$, $D'$.
e. None.
5
We follow our intuition and generalize the cases of $D_4$ and $D_3$ with no formal argumentation.
For both cases, Elements include rotations $\nfrac{i}{n} 360$ for $i = 1, 2, \dots, n-1$. Counts $n$.
Even case only. Flips about the $ith$ diagonal (counts $n/2$), and Flips about the $ith$ axis (counts $n/2$)
Odd case only. Flips about the $ith$ diagonal (counts $n$).
$D_n$ is going to have a total of $2n$ elements; This fact was mentioned in the textbook though.
11
Notation. We donate Rotation by T and Reflection by F.
Lemma. Through Caylay table in page 33, $TT = T$, $FF = T$, $TF = F$, and $FT = F$. In other words $X^2 = T$, and $XY = F$ if $X \neq Y$.
Theorem. Observe we can re-structure the given composed function as $a^2b^2b^2 ac c^2 c^2 a^2 ac = TTTacTTTac = (TTTac)^2 = T$.
Therefore, Regardless of the choices of $a, b, c$, The given function is always a rotation.
13
$D = H R_{90} = R_{90} V$.
21
$X \neq H, V, D, D', R_0, R_{180}$, As otherwise $X^2 = R_0$ and then $Y = R_{90}$. For either of the remaining two cases $X = R_{90}$ or $X = R_{270}$, Necessarily $Y = R_{270}$.