Write elements as permutations in $S_6$ of the dihedral group of symmetries of a regular hexagon inscribed in a unit circle with one vertex on the $x$-axis.
For the first part:
Let the hexagon be denoted the name $ABCDEF$, where $A$ is on the x-axis, $B$ above it and so on. I assign $1$ to $A$, $2$ to $B$ and in this way $6$ to $F$. The identity permutation is $e$, which is $R_0$. Then $R_{60}=(123456)$,$R_{120}=(135)(246)$, $R_{180}=(14)(25)(36)$, $R_{240}=(153)(264)$, $R_{300}=(165432)$, where $R_{\theta}$ denotes the rotation in the anti-clockwise direction by $\theta$.
For the reflections, there are six symmetries.
$V=(23)(14)(65)$, where $V$ is the vertical symmetry.
$H=(35)(26)$, where $H$ is the horizontal symmetry.
$D_1=(13)(46)$ , where $D_1$ is the line joining vertices $2$ and $5$
$D_2=(24)(15)$ , where $D_2$ is the line joining vertices $3$ and $6$
$D_3=(12)(36)(45)$ , where $D_3$ is the line passing through the midpoints of line joining vertices $1$,$2$ and$4$, $5$ .
$D_4=(16)(34)(25)$ , where $D_4$ is the line passing through the midpoints of line joining vertices $1$,$6$ and$4$, $3$.
Is this alright??
Thanks for the help!!
Best Answer
The symmetries of a regular hexagon are those of dihedral group 6, $D_6$.
Let $R_n$ denote a rotation around the centre of an angle $n \pi/ {\bf 3}$, for $n\in \{0,2,3,4,5\}$.
Let $r_n$ denote a reflection about a line through the centre which is at angle of $n \pi / {\bf 6}$ from the x-axis, for $n\in\{0,1,2,3,4,5\}$.
The permutations are thus: $$\begin{align} R_0 & = () \\ R_1 & = (123456) \\ R_2 & = (135)(246) \\ R_3 & = (14)(25)(36) \\ R_4 & = (153)(264) \\ R_5 & = (165432) \\[1ex] r_0 & = (26)(35) & = H \\ r_1 & = (12)(36)(45) & = D_3 \\ r_2 & = (13)(46) & = D_1 \\ r_3 & = (14)(23)(56) & = V \\ r_4 & = (15)(24) & = D_2 \\ r_5 & = (16)(25)(34) & = D_4 \end{align}$$ Tip: When written in this order a nice pattern emerges. Pick a vertex and look to where it maps to in each subsequent rotation or reflection. Use this pattern to help check your work in exams and such.