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Groups : Third Sylow Theorem

Suppose a simple group G of order 660 is a subgroup of the symmetric group S11 ( here S is with 11 as a subscript ) and x = {1,2,3,4,5,6,7,8,9,10,11}( a permutation like 1 goes to 2 and so on, I think ) in G. If P equals the span of x ( or <x> ) determine the permutations which generate NGP ( here G is written as the subscript of N (the normalizer of P in G, I think).
Hint: if you know the order of the normalizer N of <x> can you find the generators for N ?
Give a solid math argument why this is correct and please keep the answer as simple as possible at the same time.

Solution Summary

Permutations are found using the third Sylow theorem. The solution is detailed and well presented. The solution received a rating of "5" from the student who posted the question.