Reviewing some of my old questions here, I am stuck at a comment in which Prof. Holt gave me an interesting example (A small one) about non-transitive $1/2-$transitive group. Here is the link {http://math.stackexchange.com/q/138937/8581}. Of course, he noted a complete answer by giving two non-trivial of these kinds of groups.
Meanwhile, it came to my mind if we could find any fixed block in this small group, $G=\{1\}$ acting on $\Omega=\{1,2\}$ which is not an orbit? I see that we get $\Omega^G=\Omega$ in this example and so, $\Omega$ itself is a fixed block but not an orbit of the action.
In fact, $1^G=\{1\}$ and $2^G=\{2\}$. That $\Omega^G=\Omega$ is achieved, is obvious so I am trying to find any other proper subset of $\Omega$.
Can someone give me other example in which we have a fixed block in any group actions that is not an orbit? Thanks.
Let a group $G$ acts on a set $\Omega$ and $\Delta⊆\Omega$. $\Delta$ is said to be a Fixed Block of $G$ if for any $g\in G, \Delta^g=\Delta$.