I am reading Michael Artin's Algebra, encountered paragraphs( 7.3) that I don't quite understand.
so there's the citation
The elementary formula which uses the partition of $S$ into orbits to count its elements. We label the different orbits which make up $S$ in someway, say as $O_1, ..... , O_k$ then $|S| = |O_1|+.....|O_k|$, the formula hs great number of applications. Example, consider the group G of orientation-preserving symmetries of a regular dodecahedron $D$. these symmetries are $11$ rotations. it is tricky to count them without error. Consider the action of $G$ on the set $S$ of the faces $D$. the stablilizer of a face is the group of ortations by multiples of $2\pi/5$ about a perpendicular through the center of $S$. So the order of $G_s$ is $5$. There are $12$ faces, and $G$ acts transitively on them, thus $|G| = 5\times 12 = 60$ or $G$ operates trasnsitively on the vertices $v$ of $D$. There are three rotations including 1 fix a vertex, so $|G_v| = 3$. There are $20$ vertices hence $|G| =3\times 20 = 60$, which checks. There is a similar computation for edges. If $e$ is an edge then $|G_e| = 2$, so since $60 = 2\times 30$, the dodecahedron has $30$ edges.
I don't quite understand when counting using vertex what are the three rotations, and when counting using edge what is exactly $G_e$ meant for ?
let $S$ be the set of $12$ faces of the dodecahedron, and let $H$ be the stablilizer of a particular face $S$, then $H$ also fixes the face opposite to $S$, and so there are two $H$-orbits of order 1, The remaining faces make up two orbits of order $5$. In this case it reads as follows $12 = 1+1+5+5$.
I really don't understand how the order of H-orbits are derived? and infact don't even understand, what are the gemometric meaning of these H-orbits?