I started reading about permutations, cycles, disjoint cycles, decomposition,... etc but I am a bit confused about the multiplication!
To be precise, I learned how to multiply two permutations but I am wondering how can I multiply a permutation by (only) one transposition.
For example, for the permutations $\sigma_1 = (1 6 5 2 7 3 4 8)$ and $\sigma_2 = (1 5 2 4 6 8)(3 7)$, the multiplication of them equals $(1 2 8 6)(3)(4 5 7) = (1 2 8 6)(4 5 7)$.
I know that $\sigma_1 = (1 6 5 2 7 3 4 8)$ consists of these transpositions $(1 8)(1 4)(1 3)(1 7)(1 2)(1 5)(1 6)$ or $(1 6)(6 5)(5 2)(2 7)(7 3)(3 4)(4 8)$. Also, it is known that multiplying a permutation by a transposition produces one more(or one fewer) number of cycles than the permutation(i.e than the number of cycles in the permutation) but I can't do this multiplication! like to multiply $\sigma_1$ by any one of it's transpositions "let it be $(4 8)$".
So, $(1 6 5 2 7 3 4 8) * (4 8) = ?$
and in general $(1 6 5 2 7 3 4 8) * (a b) = ?$
Thanks for any example or a reference clarifies the procedure
We have $(16527348)(48) = (4165273)$. Start with the first number of the cycle on the right ($4$ here). Now by the first cycle it gets send to $8$ and the number $8$ gets send to $1$ therefore we get $(41...$. Now we need to know where $1$ is getting send to. Therefore we start on the right and go through the permutations again. Do this until all numbers were considered.