Can anyone help me in this proof ?

If A and B are in S_n, then is the permutation that has the same cycle structure as B and that is obtained by applying A to the symbols in B.

Printable View

- Sep 21st 2008, 08:11 PMdeniselim17need help in this permutation proof
Can anyone help me in this proof ?

If A and B are in S_n, then is the permutation that has the same cycle structure as B and that is obtained by applying A to the symbols in B. - Sep 21st 2008, 10:08 PMwisterville
Hello,

Prove it when the length of A is 2 and the B is cyclic:

A=(i j), B=(m_1 m_2... m_k).

Consider several cases depending on whether i, j are included in B.

For example, if i=m_n and j doesn't occur in B,

ABA^{-1}=(m_1 m_2... m_{n-1} j m_{n+1} ... m_k).

Bye. - Sep 22nd 2008, 10:09 AMThePerfectHacker
Let (where ). Let be a permutation. Prove that . Now let be any permuation. Then we can write where are disjoint cycles. Therefore, . But each has the same structure as by above. Therefore, and both have the same number of disjoint cycles in their cyclic decompositition. Therefore and have the same structrure.