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Groups and FieldsHardThe Sylow Theorems

The Sylow Theorems · Hard problem

S5S_5 has order 120=23⋅3⋅5120=2^3\cdot3\cdot5. The number of 55-cycles in S5S_5 is 4!=244!=24. How many Sylow 55-subgroups does S5S_5 have?