Understanding 2-Transitivity in Multiply Transitive Groups

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In summary, the conversation discusses a problem involving the set Z and its partitions into two sets of three. The set X is labeled with 10 partitions and represented as permutations in S6. The first question asks to show that (S6)^ is 2-transitive on X using (abc)^ and (def)^. The second question asks for the number of elements in (S6)^ that fix both 0 and 1, and concludes that (S6)^ is not 3-transitive on X. The speaker thanks the group and mentions that they have already solved the problem.
  • #1
laptopmarch
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Hi All,

I have a hard time answering the following. I need some help.

Let Z={a,b,c,d,e,f} and let X denote the set of 10 partitions of Z into two sets of three. Label the members of X as follows:

0 abc|def
1 abd|cef
2 abe|cdf
3 abf|cde
4 acd|bef
5 ace|bdf
6 acf|bde
7 ade|bcf
8 adf|bce
9 aef|bcd

Let g->g^ denote the representation of S6=Sym(Z) as permutations of X.

1. By considering (abc)^ and (def)^, show that (S6)^ is 2-transitive on X.

2. How many elements of (S6)^ fix both 0 and 1? Find them. Deduce that (S6)^ is not 3-transitive on X.

Thank you very much. :)
 
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  • #2
I have already solved the problem.thank you.
 

1. What are Multiply Transitive Groups?

Multiply Transitive Groups are groups that act transitively on two or more sets of elements. This means that for any two pairs of elements in each set, there exists a group element that maps one pair to the other.

2. How are Multiply Transitive Groups different from other groups?

Multiply Transitive Groups are unique in that they have a higher level of symmetry and flexibility compared to other groups. This means that they have more possible actions and transformations on sets of elements.

3. What are some examples of Multiply Transitive Groups?

Some examples of Multiply Transitive Groups include the symmetric group, the alternating group, and the dihedral group. These groups have a wide range of applications in fields such as mathematics, physics, and computer science.

4. What are the practical uses of Multiply Transitive Groups?

Multiply Transitive Groups have many practical uses in various fields. For example, they are used in cryptography to create secure codes and in coding theory to design error-correcting codes. They also have applications in crystallography, chemistry, and computer graphics.

5. What are some challenges in studying Multiply Transitive Groups?

One of the main challenges in studying Multiply Transitive Groups is their complexity. As they have a high level of symmetry and many possible actions, it can be difficult to fully understand their properties and behaviors. Additionally, finding efficient algorithms for manipulating these groups is an ongoing research topic.

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