Determination of multiplication table of symmetric group:S3

The purpose of this survey article is to give a detailed description of the construction of the multiplication table of symmetric group:S3. The survey article is meant as a walk-through of the entire process rather than a summary and is written for people relatively new to group theory. If you're looking for terse summary descriptions, check out symmetric group:S3 and element structure of symmetric group:S3.

The final table

We first provide the final multiplication table in cycle decomposition notation and also one-line notation for the group $S_3$ acting on the set $\{1,2,3\}$. The convention followed here is that the column element is on the right and the row element is on the left, and functions act on the left. Hence, to determine the effect of the composite permutation on any element of $\{1,2,3\}$, we must first apply the permutation given by the column element and then apply the permutation given by the row element.