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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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# Multiply Transitive Groups

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