Confusion on how groups acts on objects

Question

im trying to learn group theory to understand particle physics and im currently reading: "A Simple Introduction to Particle Physics ,Part I - Foundations and the Standard Model". In the chapter called Group Actions he uses an example of three colorred eggs (ROY) and defines the actions:

1. Let e be doing nothing to the set, so e(ROY ) = (ROY ).
2. Let g1 be a cyclic permutation of the three, g1(ROY ) = (OY R)
3. Let g2 be a cyclic permutation in the other direction, g2(ROY ) = (Y RO)
4. Let g3 be swapping the first and second, g3(ROY ) = (ORY )
5. Let g4 be swapping the first and third, g4(ROY ) = (Y OR)
6. Let g5 be swapping the second and third, g5(ROY ) = (RY O)

which gives us this table:

$ \begin{array}{|c||c|c|c|c|c|c|} \hline(G, \star) & e & g_{1} & g_{2} & g_{3} & g_{4} & g_{5} \\ \hline e & e & g_{1} & g_{2} & g_{3} & g_{4} & g_{5} \\ \hline g_{1} & g_{1} & g_{2} & e & g_{5} & g_{3} & g_{4} \\ \hline g_{2} & g_{2} & e & g_{1} & g_{4} & g_{5} & g_{3} \\ \hline g_{3} & g_{3} & g_{4} & g_{5} & e & g_{1} & g_{2} \\ \hline g_{4} & g_{4} & g_{5} & g_{3} & g_{2} & e & g_{1} \\ \hline g_{5} & g_{5} & g_{3} & g_{4} & g_{1} & g_{2} & e \\ \hline \end{array} $

Later in the paper he generalizes the three eggs by a set of objects $M= {m_0,m_1,m_2}$ and then building an algebra with these objects the orthonormal vectors are: $m_0 \rightarrow |m_0> $ and so on. During the example of the section on how $S_3$ acts on theses objects he writes:

$ g_{1}\left(c_{0}\left|m_{0}\right\rangle+c_{1}\left|m_{1}\right\rangle+c_{2}\left|m_{2}\right\rangle\right)=\left(c_{0}\left|g_{1} m_{0}\right\rangle+c_{1}\left|g_{1} m_{1}\right\rangle+c_{2}\left|g_{1} m_{2}\right\rangle\right) $ and from the multiplication table we can see that $ g_1m_0=m_1 , g_1m_1=m_0,g_1m_2=m_2, \Rightarrow \left(c_{0}\left|g_{1} m_{0}\right\rangle+c_{1}\left|g_{1} m_{1}\right\rangle+c_{2}\left|g_{1} m_{2}\right\rangle\right)=\left(c_{0}\left|m_{1}\right\rangle+c_{1}\left| m_{0}\right\rangle+c_{2}\left|m_{2}\right\rangle\right) $

I have two questions about all of this. First of all im looking at the table and i don't understand how the last relations came to be. To my understanding $g_1m_1=m_2$. My second question is this: How are the eggs and the $m$ vectors related? What is the analogy here? Is $|m_1>$ the red egg for example? If so why aren't their positions swapped in the same way in how it swaps the eggs?

Answer 1

Maybe I'm wrong but if I did it correctly I found that from the table we have:

$g_1=(132),\ g_2=(123),\ g_3=(23),\ g_4=(12),\ g_5=(13)$

Which has something problematic since then $g_1$ couldn't stabilize $m_2$.