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altcmdesc
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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
Thanks
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.
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?
That's one S_n action.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 of them.altcmdesc said:It acts on itself by conjugation, right?
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 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.
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.
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.
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.
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.