symmetric group on three letters
This example is of the symmetric group on letters, usually denoted by . Here, we are considering the set of bijective functions on the set which naturally arise as the set of permutations on . Our binary operation is function composition which results in a new bijective function. This example develops the table for . We start by listing the elements of our group. These elements are listed according to the second method as described in the entry on permutation notation.
Here, our group is just . Now we can start to multiply and then fill in the table. First, we calculate the square of each element.
Next, we will fill in the upper right block, we only need and since we can use the fact that there can be no repetition in any row or column.
The other blocks are also similar. Now continuing with the upper left 3 x 3 block, we go through the table again using the fact that there can be no repetition in any row or column.
Similarly, we complete the final blocks of the table.
Finally, we fill in the table using the calculated values above.