I'm reading Beardon's, Algebra and Geometry. Here:
I'm having a little trouble understanding the marked part. In the expression $\rho_j(x)=\rho(x)$, who is $\rho_j(x)$? Is it one cycle that is equal to the entire permutation? And in $\rho_i(x)=x$, is it one (trivial?) cycle that fixes each integer in $\rho_i$? I'm a little confused at:
If they are actually what I think they are;
Assuming they are, why are they used? I guess that the author wants to use $\rho_i$ to represent the identity of a particular cycle and $\rho_j$ to represent the entire permutation as one element of the factorization. Is that it?