# S_n acting on a set with more than n elements

1. Nov 30, 2009

### altcmdesc

I'm wondering how the group S_n can act on a set with more than n elements? 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.

Thanks

2. Nov 30, 2009

### latentcorpse

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. Nov 30, 2009

### Hurkyl

Staff Emeritus
Actions can have more than one orbit.

4. Nov 30, 2009

### latentcorpse

i don't follow...

5. Nov 30, 2009

### Hurkyl

Staff Emeritus
Oh, by the way....

There is a rather obvious action of S_n on a particular set of n! elements....

6. Dec 1, 2009

### altcmdesc

So, essentially, S_n can act on any set of more than n elements by just permuting n of the elements and leaving the rest alone?

7. Dec 1, 2009

### Hurkyl

Staff Emeritus
That's one S_n action.

However, what if you have two sets of n elements arranged into a set of 2n elements? What can you do with that?

But the hint I was trying to give earlier is that S_n has a very natural action on itself, viewed as a set of n! elements. (Can you see what that action would be?)

8. Dec 1, 2009

### altcmdesc

It acts on itself by conjugation, right? Does that action apply to sets with less than n! elements as well by taking an element $$a$$ in the finite set and multiplying it by $$pap^{-1}$$?

9. Dec 1, 2009

### Hurkyl

Staff Emeritus
That's one of them.

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)