Let $X=\{n \in \mathbb{N}; 1 \leq n \leq 9\}$. Let $\sigma= (123)(456)(789)$ and $\tau=(12)(34)(56)(78)$, and $G = <\sigma,\tau>$. $X$ is a $G$-set (G acts on X).
a) What is the orbit of $G$ acting on 2? What about 6? What about 8?
b) Express $|G_1|$ via $|G|$.
My solution:
I know that the orbit of an element is the set $Gx = \{gx ; g \in G\}$ therefore, we can calculate $G = \{ id, \sigma, \tau, \sigma \tau, \tau \sigma,\sigma^2 \tau , \sigma^2,\tau \sigma^2\}$ since the order of $\sigma$ is 3 and the order of $\tau$ is 2 (lcm of the length of their disjoint cycles).
Then I calculated $G = \{id, (123)(456)(789), (12)(34)(56)(78), (1354), (2463)(89),(2364)(89),(132)(465)(798),(1453)(65)(79)\} $ So $G2=\{2,3,1,4,3,1,2\}=\{1,2,3,4\}$. Is this correct? If yes, I will continue to do 6 and 8 similarly.
Using formula $|G|=|G1||G_1|$ I get that $|G_1|=|G| / |G1| = 8/2=4$. I don't know if this is okay...