Question:
The number of elements of order two in the group \(S_4\) is equal to _________.
The number of elements of order two in the group \(S_4\) is equal to _________.
Show Hint
In symmetric groups, an element’s order equals the least common multiple (LCM)
of the lengths of its disjoint cycles.
Updated On: Dec 6, 2025
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Correct Answer: 9
Solution and Explanation
Step 1: Possible elements of order 2 in \(S_4\).
In the symmetric group \(S_4\), an element has order 2 if it is a product of disjoint transpositions. The possible cycle structures for elements of order 2 are: - A single transposition (2-cycle), e.g., \((1\ 2)\). - A product of two disjoint transpositions, e.g., \((1\ 2)(3\ 4)\).
Step 2: Count each type.
- Number of single transpositions: \(\binom{4}{2} = 6.\) - Number of disjoint 2-cycles: Choose 4 distinct elements and pair them up. The number of such elements is \[ \frac{1}{2}\binom{4}{2} = 3. \]
Step 3: Add totals.
\[ 6 + 3 = 9. \]
Step 4: Conclusion.
Hence, the number of elements of order two in \(S_4\) is \(\boxed{9}.\)
In the symmetric group \(S_4\), an element has order 2 if it is a product of disjoint transpositions. The possible cycle structures for elements of order 2 are: - A single transposition (2-cycle), e.g., \((1\ 2)\). - A product of two disjoint transpositions, e.g., \((1\ 2)(3\ 4)\).
Step 2: Count each type.
- Number of single transpositions: \(\binom{4}{2} = 6.\) - Number of disjoint 2-cycles: Choose 4 distinct elements and pair them up. The number of such elements is \[ \frac{1}{2}\binom{4}{2} = 3. \]
Step 3: Add totals.
\[ 6 + 3 = 9. \]
Step 4: Conclusion.
Hence, the number of elements of order two in \(S_4\) is \(\boxed{9}.\)
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