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Thanks

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Thanks

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S2 can act on {1,2,3} to give {2,1,3} doesn't it??? *hopes not to have just made a fatal error?*

- #3

Hurkyl

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Actions can have more than one orbit.I'm basically only looking for some idea as to how to get started on describing such an action and how to think of that action.

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i don't follow...

- #5

Hurkyl

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There is a rather obvious action of S_n on a particular set of n! elements....I'm wondering how the group S_n can act on a set with more than n elements?

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- #7

Hurkyl

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That's one S_n action.

However, what if you have

But the hint I was trying to give earlier is that S_n has a very natural action

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- #9

Hurkyl

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That's one of them.It acts on itself by conjugation, right?

There's another one that, in some sense, is more fundamental: it acts by multiplication.

It turns out that every transitive action of a group G is isomorphic to a quotient of this one -- that is, G acting on the set G/H by multiplication, where H is a (not necessarily normal) subgroup.

And every action of a group G is a disjoint union of transitive actions -- it's the union of its orbits, and each orbit is a transitive G-set.

Multiplying on the right isn't defined. (although there are generalizations where right and left multiplication can happen to your set)[tex]a[/tex] in the finite set and multiplying it by [tex]pap^{-1}[/tex]?

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