Is there a difference between $(2\,\,4) \circ (1\,\,3)$ and $(2\,\,4)(1\,\,3)$?

Question

Is there a difference between $(2\,\,4) \circ (1\,\,3)$ and $(2\,\,4)(1\,\,3)$?

Answer 1

Here $(24)(13)$ refers to the permutation \begin{align} \begin{pmatrix} 1 & 2 & 3 & 4\\ 3 & 4 & 1 & 2 \end{pmatrix}, \end{align} in two-line notation, whereas $(24)$ and $(13)$ are the permutations \begin{align} \begin{pmatrix} 1 & 2 & 3 & 4\\ 1 & 4 & 3 & 2 \end{pmatrix} \ \ \text{ and } \ \ \begin{pmatrix} 1 & 2 & 3 & 4\\ 3 & 2 & 1 & 4 \end{pmatrix} \end{align} respectively. When we compose $(24)$ with $(13)$ to get $(24)\circ (13)$ we mean \begin{align} \begin{pmatrix} 1 & 2 & 3 & 4\\ 3 & 2 & 1 & 4\\ 3 & 4 & 1 & 2 \end{pmatrix} \ \ \text{which simplifies to } \ \ \begin{pmatrix} 1 & 2 & 3 & 4\\ 3 & 4 & 1 & 2 \end{pmatrix}. \end{align} Hence the composition has the same two-line expression as $(24)(13)$ so $(24)\circ(13) = (24)(13)$.

Answer 2

$(2\,\,4)(1\,\,3)$ is a misuse of the multiplication record instead of composing. We have the same example in matrix multiplication. In fact, it is not about multiplication of matrices, but about composing a matrix. So $A B$ is actualy a matrix composition, not a matrix multiplication.

Answer 3

The first one is the result of $(2\, 4)$ on the left side and $(1\, 3)$ on the right side of the binary operation $\circ$, commonly understood to mean the composition of the two permutations as functions (unless stated otherwise); the second is the concatenation of $(2\, 4)$ and $(1\, 3)$ and is commonly understood to mean the same thing (when the context is clear).