Discussion
In the group \(S_{3}\), there are exactly :
Answer: B
\(S_{3}\) has 3 elements with \(x^{3} = e\) and 4 elements with \(x^{2} = e\).
\(S_{3}\) has six elements: the identity, three transpositions (order 2) and two 3-cycles (order 3).
- \(x^{3} = e\): the identity and the two 3-cycles, 3 elements.
- \(x^{2} = e\): the identity and the three transpositions, 4 elements.
Options with 2 elements for \(x^{3} = e\), or 3 for \(x^{2} = e\), forget to count the identity, which satisfies every equation \(x^{k} = e\).
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