Find All Transitive G-Sets Up to Isomorphism w/ Subgroups of G

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In summary: G/H.In summary, the conversation discusses the relationship between transitive G-sets and coset spaces of subgroups of G. It is stated that any transitive G-set is isomorphic to the coset space of some subgroup of G. This means that all transitive G-sets can be determined up to isomorphism by finding all subgroups of G. The example of G = S_3 is used to illustrate this concept. It is also noted that the coset space of a subgroup may not always be a quotient group if the subgroup is not normal.
  • #1
daveyinaz
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I read in a book on groups and representations that any transitive [tex]G[/tex]-set is isomorphic to the coset space of some subgroup of [tex]G[/tex].
Does this mean we can determine all transitive [tex]G[/tex]-sets up to isomorphism simply by finding all subgroups of [tex]G[/tex]?

Just want to make sure that if this is the case that I have in my mind the right idea, so we take [tex]G = S_3[/tex], then all transitive [tex]G[/tex]-sets are up to isomorphism...
[tex]G / \{e\}, G / \langle (12) \rangle , G / \langle (132) \rangle , G / G[/tex]?

Note I do realize that [tex]\langle (13) \rangle[/tex] is also a subgroup of [tex]S_3[/tex] but the way I see it the coset space would be the same as [tex]G / \langle (12) \rangle[/tex]
 
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  • #2
That's correct, except that the coset space of a subgroup of G is not in general a quotient group (since the subgroup may not be normal)
 
  • #3
Well that would make sense right? Since a G-set isn't necessarily a group...and if it's isomorphic to some coset space, then that coset space isn't a group either.
 
  • #4
Yup, but writing

[tex]
G / \{e\}, G / \langle (12) \rangle , G / \langle (132) \rangle , G / G
[/tex]

This is notation for quotient groups. If H is a subgroup of G, the set of cosets is often denoted cos(G:H)
 

1. What is a transitive G-set?

A transitive G-set is a set with a group action of G, where for any two elements in the set, there exists an element in G that maps one element to the other. In other words, the group G acts transitively on the set.

2. What is an isomorphism in the context of G-sets?

An isomorphism in the context of G-sets is a bijective mapping between two G-sets that preserves the group action. In other words, the two G-sets are essentially the same, just with different labels or representations.

3. How do you find all transitive G-sets up to isomorphism?

To find all transitive G-sets up to isomorphism, you first need to find all possible subgroups of G. Then, for each subgroup, you need to find all possible cosets. These cosets, along with the group action of G, will determine a unique transitive G-set. By finding all possible combinations of subgroups and cosets, you can find all transitive G-sets up to isomorphism.

4. What is the significance of finding all transitive G-sets up to isomorphism?

Finding all transitive G-sets up to isomorphism is significant because it can provide a complete understanding of the group structure of G. It also allows for the identification of common patterns and symmetries, which can aid in further research and applications of the group.

5. What are the applications of finding all transitive G-sets up to isomorphism?

The applications of finding all transitive G-sets up to isomorphism include cryptography, coding theory, and group theory. It also has applications in computer science, particularly in areas such as network routing and database design. Additionally, understanding the group structure of G can have implications in physics, chemistry, and other scientific fields.

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