Group Actions on Sets: Understanding the Permutation Group S_3

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In summary, the conversation is discussing the concept of group actions and homomorphisms, specifically in regards to the group S_3 and how it can act on a set X with 4 elements and the different ways it can embed into the group S_4 in a homomorphic way. The conversation also touches on the idea of using generators and the possible existence of other homomorphisms.
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T-O7
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Okay, so I'm trying to understand the notion of group actions, and I'm having a little difficulty understanding how to work on this question:

Describe all the ways the group [tex]S_3[/tex] can act on a set [tex]X[/tex]with 4 elements.

I mean, an action assigns with every element in [tex]S_3[/tex] a permutation of the set X. The confusing thing for me now is that the group we start with is a permutation group itself, so it's like for every permutation in [tex]S_3[/tex], we assign a permutation of X. But how does that help me answer the question? :confused:
 
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You are asked to describe all homomorphisms from S_3 to S_4.

One way to do this is to pick generators of S_3 in a suitable fashion.

We may use the fact it is generated by transpositions.

S_3 is generated by (12) and (23)

How cany you embed these elements in S_4 in a group homomorphic way?

Hint: if phi is a homomorphism, ord(phi(x)) divides ord(x).
 
  • #3
Okay, so describing all homomorphisms from [tex]S_3[/tex] to [tex]S_4[/tex] seems a little more tangible. I suppose there are four "natural" homomorphisms, described by simply ignoring one element in [tex]S_4[/tex], and using the permutation of [tex]S_3[/tex] to permute the remaining 3 elements. Hmm...by the looks of it, I don't think there can be any other homomorphism, but I'm now thinking of a way to show that.
 
  • #4
Well, there are other homomophisms; who said the map needed to be injective?
 

FAQ: Group Actions on Sets: Understanding the Permutation Group S_3

What is a group action/operation?

A group action/operation is a mathematical concept that describes how a group (a set with a binary operation that satisfies certain properties) acts on another set. It is a way of combining elements from both sets to produce a new element in the second set.

What are the properties of a group action/operation?

The properties of a group action/operation include: closure, meaning the result of the operation must be an element of the second set; associativity, meaning the order in which the elements are combined does not matter; identity, meaning there is an element in the group that acts as a neutral element; and invertibility, meaning every element in the second set has an inverse element in the group.

How is a group action/operation represented?

A group action/operation is typically represented using a table, known as a Cayley table, that shows the result of combining each element from the group with each element from the second set. It can also be represented using a function notation, such as g(x), where g is an element from the group and x is an element from the second set.

What is the significance of group actions/operations in mathematics?

Group actions/operations have many applications in mathematics, including in abstract algebra, geometry, and topology. They help to understand and solve problems related to symmetry, transformation, and structure in various mathematical objects.

What are some examples of group actions/operations?

Some examples of group actions/operations include: rotations and reflections of a geometric shape, multiplication and addition of matrices, permutations of a set, and automorphisms of a mathematical structure. Group actions/operations can also be used to describe physical transformations, such as translations and rotations in physics.

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