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What can we say about the set that a group can act on it as a permutation group? My question is about the structure of elements of a group G when acts on an arbitrary set Ω as a permutation group. For example the alternating group $A_4$ has different elements structures in its different actions(every element in its action on 6 points has two cycles of sizes 2 or two cycles of sizes 3). What is important for me is the structure of elements in their disjoint cycle decomposition.

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1 Answer 1

If the (finite) group $G$ acts on a set $\Omega$, then $\Omega$ is the disjoint union of the orbits.

On each orbit $G$ acts transitively. Now if $G$ acts transitively on a set $\Delta$, then the action of $G$ on $\Delta$ is similar to the action of $G$ on the cosets of the stabilizer $G_{\alpha}$ of an element $\alpha \in \Delta$.

That is, every transitive action is similar to the action on the coset space $G/H$, for some $H \le G$. (It is enough to pick a representative $H$ for each conjugacy class of subgroups, because similarity of the actions on $G/H$ and $G/K$ corresponds to $H$ and $K$ being conjugate.)

This gives a framework to answer your questions: start with these transitive actions $G/H$ and put them together by disjoint union in any way you want.

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Thanks. So what you said is that there are 2 actions for a finite group? –  M. R Mar 5 '13 at 10:18
Why two? I don't understand. For instance $S_{3}$ has $4$ conjugacy classes of subgroups, and thus four pairwise non-similar transitive actions. And then you can put these together to get many more (non-transitive) actions. –  Andreas Caranti Mar 5 '13 at 10:22
Thank you Professor –  M. R Mar 5 '13 at 10:46
@MehdiRezaei, you're very welcome! –  Andreas Caranti Mar 5 '13 at 10:46
what did you mean of "conjugacy classes of subgroups"? –  M. R Dec 9 '13 at 11:13

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