Proving Graph Theory with Group Permutations | G = Sn and S Set

In summary, the conversation discusses the definition of a permutation on a set and how it acts as a bijection. The speaker is unsure about a possible typo in the problem and is seeking clarification on how the function acts on the set. The other speaker explains that a permutation on a finite set is a function that is a bijection, using the example of a set of {1,2,3}.
  • #1
TheMathNoob
189
4

Homework Statement


The problem is attached. I don't get this part. Let G = Sn be the group of all permutations of S. S is a set, so how can we permute something in a set?. Neither I know if the 4 power in the S is a typo.

Homework Equations

The Attempt at a Solution

 

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  • #2
A permutation is just a function from S to S which is a bijection. I agree the 4 looks like a typo, or perhaps is referencing a footnote?
 
  • #3
Office_Shredder said:
A permutation is just a function from S to S which is a bijection. I agree the 4 looks like a typo, or perhaps is referencing a footnote?
Yes, I have to prove that it is a bijection, but I don't understand how this function acts on the set because as I said, you can't permute things in a set.
 
  • #4
You have to prove that [itex]\sigma[/itex] acts as a bijection on the vertices of G. I am telling you that the definition of a permutation on a finite set (how [itex]\sigma[/itex] acts on S, not V) is a function that is a bijection.

For example, suppose that the set is {1,2,3}. Then one permutation f(n) might be f(1) = 2, f(2) = 3, f(3) = 1. Another might be f(1) = 1, f(2) = 3, f(3) = 2.
 

Related to Proving Graph Theory with Group Permutations | G = Sn and S Set

1. What is Graph Theory?

Graph Theory is a branch of mathematics that deals with the study of graphs, which are mathematical structures used to model pairwise relations between objects.

2. What are Group Permutations?

Group Permutations are mathematical structures that represent a set of objects and the various ways in which the objects can be rearranged or combined. In other words, they are used to describe symmetries in a set of objects.

3. How is Graph Theory related to Group Permutations?

Graph Theory and Group Permutations are closely related because graphs can be represented as permutations of a set of objects. This allows for the application of group theory to solve problems in graph theory.

4. What is the significance of using G = Sn and S Set in proving Graph Theory?

G = Sn and S Set are specific notations used in group theory to represent symmetric groups, which are used to model permutations of a set of objects. By using these notations, we can apply the principles of group theory to prove various concepts in graph theory.

5. How can Proving Graph Theory with Group Permutations be useful?

Proving Graph Theory with Group Permutations can be useful in understanding the underlying mathematical structures and symmetries in graphs, which can help in solving complex problems and developing new algorithms for graph-based applications.

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