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altcmdesc

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Thanks

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In summary, the group S_n can act on a set with more than n elements by conjugating it and then multiplying by pap^{-1}.

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altcmdesc

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Thanks

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latentcorpse

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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.altcmdesc said: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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latentcorpse

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i don't follow...

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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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altcmdesc

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Hurkyl

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That's one S_n action.altcmdesc said:

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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altcmdesc

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Hurkyl

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That's one of them.altcmdesc said: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]?

A permutation group, denoted as S_{n}, is a group of all possible permutations of a set with n elements. In other words, it is a group of all bijective functions from a set to itself.

When S_{n} acts on a set with more than n elements, it acts as a group of bijections, meaning it rearranges the elements of the set in all possible ways while keeping the same number of elements.

No, S_{n} cannot act on a set with less than n elements because it would not be a bijection. In order for S_{n} to act on a set, the set must have the same number of elements as the order of the group.

The order of S_{n} when it acts on a set with more than n elements is n!, which is the number of possible permutations of n elements. This is because each permutation is a distinct element in the group.

S_{n} is a specific type of symmetric group, known as the full symmetric group. It is the symmetric group of degree n, meaning it consists of all possible permutations of n elements. Other symmetric groups, such as alternating groups, are subgroups of S_{n}.

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