The number of 5-Sylow subgroups in the symmetric group \( S_5 \) of degree 5 is \( \underline{\hspace{1cm}}\).
The number of 5-Sylow subgroups in the symmetric group \( S_5 \) of degree 5 is \( \underline{\hspace{1cm}}\).
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Correct Answer: 6
Solution and Explanation
- The number of Sylow 5-subgroups must divide \( |S_5| = 5! = 120 \).
- The number must also be congruent to 1 modulo 5.
Thus, the possible values are divisors of 120 that are congruent to 1 modulo 5. The only such divisor is 6. Hence, the number of 5-Sylow subgroups in \( S_5 \) is \( \boxed{6} \).
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