This question from group theory asks for the size (or order) of a specific symmetric group, \(S_3\).
Step 1: Understanding the Question:
We need to find the total number of elements in the group \(S_3\). The group \(S_3\) is the set of all possible permutations (bijective functions) of a set with 3 distinct elements, for example, the set \(\{1, 2, 3\}\).
Step 2: Key Formula or Approach:
The order of the symmetric group on \(n\) elements, denoted \(|S_n|\), is given by the factorial of \(n\).
\[ |S_n| = n! \]
Step 3: Detailed Explanation:
A permutation of \(\{1, 2, 3\}\) is simply a reordering of these three numbers. We can count the number of possible orderings directly:
There are 3 choices for the first position.
After choosing the first, there are 2 choices remaining for the second position.
After choosing the first two, there is only 1 choice left for the third position.
The total number of permutations is the product of these choices: \(3 \times 2 \times 1\).
Using the formula from Step 2 for \(n=3\):
\[ |S_3| = 3! = 3 \times 2 \times 1 = 6 \]
The 6 elements are: the identity, three transpositions ((1 2), (1 3), (2 3)), and two 3-cycles ((1 2 3), (1 3 2)).
Step 4: Final Answer:
The order of the group \(S_3\) is 6.