S_n acting on a set with more than n elements

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}.
  • #1
altcmdesc
66
0
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
 
Physics news on Phys.org
  • #2
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
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.
Actions can have more than one orbit.
 
  • #4
i don't follow...
 
  • #5
Oh, by the way...

I'm wondering how the group S_n can act on a set with more than n elements?
There is a rather obvious action of S_n on a particular set of n! elements... :wink:
 
  • #6
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
altcmdesc said:
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?
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
It acts on itself by conjugation, right? Does that action apply to sets with less than n! elements as well by taking an element [tex]a[/tex] in the finite set and multiplying it by [tex]pap^{-1}[/tex]?
 
  • #9
altcmdesc said:
It acts on itself by conjugation, right?
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.


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

1. What is the definition of a permutation group?

A permutation group, denoted as Sn, 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.

2. How does Sn act on a set with more than n elements?

When Sn 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.

3. Can Sn act on a set with less than n elements?

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

4. What is the order of Sn when it acts on a set with more than n elements?

The order of Sn 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.

5. How is Sn related to symmetric groups?

Sn 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 Sn.

Similar threads

  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
2
Replies
36
Views
2K
  • Calculus and Beyond Homework Help
Replies
28
Views
1K
  • Calculus and Beyond Homework Help
Replies
11
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
995
  • Linear and Abstract Algebra
Replies
15
Views
4K
  • Calculus and Beyond Homework Help
Replies
24
Views
2K
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
351
  • Calculus and Beyond Homework Help
Replies
1
Views
758
Back
Top