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.
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.
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.