Understanding 2-Transitivity in Multiply Transitive Groups

[laptopmarch]

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. :)

 

[laptopmarch]

I have already solved the problem.thank you.