Question

The number of elements in the set \[ \{ x \in S_3 : x^4 = e \}, \quad \text{where} \quad e \text{ is the identity element of the permutation group } S_3, \text{ is} ............... \]

Hint:

To find the elements of a group that satisfy a condition like \( x^4 = e \), consider the order of each element.

Answer 1

Correct Answer: 4

Step 1: Understand the elements of \( S_3 \).
The group \( S_3 \) consists of all permutations of 3 elements, and has 6 elements: \[ S_3 = \{ e, (12), (13), (23), (123), (132) \}. \] We are looking for elements \( x \in S_3 \) such that \( x^4 = e \). This means the order of \( x \) must divide 4.
Step 2: Analyze the orders of the elements in \( S_3 \).
- The identity element \( e \) has order 1. - The transpositions \( (12), (13), (23) \) have order 2. - The 3-cycles \( (123), (132) \) have order 3. None of the elements in \( S_3 \) have order 4, but the identity element satisfies \( e^4 = e \), and each transposition satisfies \( x^4 = e \).
Step 3: Conclusion.
Thus, the elements that satisfy \( x^4 = e \) are \( e \), and the transpositions \( (12), (13), (23) \), giving a total of 4 elements. The number of elements is: \[ \boxed{4}. \]