Prove that the number of cycles in a permutation is always changed by exactly 1 when you multiply it by a transposition! more precisely if $c(π)$ denote the number of cycle in the permutation $π$ and if $τ = (x y)$ is a transposition you are to prove that
• $c( πτ)=c(π)+1$ if $x$ and $y$ belong to same cycle in $π$
• $c( πτ)=c(π)+1$ if $x$ and $y$ belong to different cycles in $π$.
After that, show that the shortest factorization of a permutation has length $n-c(π)$.
anyone help me how to prove last part using two results .
