confusion over transpositions and cycle notation Suppose we have a permutation on the set {1,2,3,4,5} and we express it in the cycle notation as (2,5,3). I interpret this to mean that every time we apply the permutation, 2 gets sent to 5, 5 gets sent to 3 and 3 gets sent to 2. All other elements are mapped to themselves. Formally, I think it is equivalent to $\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 5 & 2 & 4 & 3 \end{pmatrix}$. An equivalent expression is (2,3)(2,5). But if I were to have a consistent interpretation of the cycle notation, then this would mean 2 gets sent to 5 first, then 5 gets sent to itself. How is it indicated that 5 gets sent to 3? 
Why am I being downvoted?
 A: Cycle notation is read right to left; in your example, we will denote $\sigma=(2 3)(2 5)$. To see where $\sigma$ maps 2, we start in the right cycle and see 2 maps to 5. Since there is no 5 in the left cycle, we stop. Thus $\sigma$ maps 2 to 5. To see where $\sigma$ takes 5, start in the right cycle again; 5 maps to 2. Now the left cycle maps 2 to 3. The composite of these two give $\sigma$ mapping 5 to 3. Finally, we see that since there is no 3 in the right cycle, $\sigma$ maps 3 to 2. Thus, we have $\sigma$ takes 2 to 5, 5 to 3, and 3 to 2. In other words, $\sigma=(253)$.
A: Permutations of $\{1, ..., 5\}$ are bijective functions from $\{1, ..., 5\} \rightarrow \{1, ..., 5\}$. $(2 \ 3)(2 \ 5)$ is the composition of the the function $(2 \ 5)$ followed by $(2 \ 3)$. The function $(2 \ 5)$ sends $2$ to $5$. Then function $(2 \ 3)$ sends $5$ to $5$. Hence the composition $(2 \ 3)\circ (2 \ 5) = (2 \ 3)(2 \ 5)$ send $2$ to $5$. Think of these as composition of functions, you can now do the same thing to all the other elements of $\{1, ..., 5\}$ and check that indeed $(2 \ 3)(2 \ 5) = (2 \ 5 \ 3)$. 
