Multiplication table for a frieze group.

Question

I am doing the following question:

I wrote the formulae for $T_n$ as $(x+2n, y)$. I'm not sure about the forumlae for $R_n$ though. I think it will be $(something,-y)$ but I am unsure about the x component here.

Also, when computing the multiplication table, is the following correct?

$T_n$ $T_n$ gives $T_{2n}$.

$R_nR_n$ gives $I$.

I am unsure what $T_nR_n$ and $R_nT_n$ would give however. Thanks for any help.

Answer 1

\begin{array}{ccc} & T_n & R_n \\ T_m & T_{m+n} & R_{m+n} \\ R_m & R_{m-n} & T_{m-n} \\ \end{array}