A group $G$ acts transitively on a non empty $G$-set $S$ if, for all $s_1,s_2 \in S$, there exists an element $g \in G$ such that $gs_1=s_2$. Characterize transitive $G$-set actions in terms of orbits. Prove your answer.
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By the definition you gave, it seems like if we choose and fix an element $\,s\in S\,$, then $\,\forall\,s_1\in S\,\exists\,g\in G\,\,s.t.\,\,gs=s_1\,$ , and since the orbit of $\,s\,$ is defined to be $\,\mathcal Orb(s):=\{gs\;:\;g\in G\}\,$, then..how many $\,G-\,$orbits are there? |
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