Consider the symmetric group Sy of degree 7. (i) Give an example of a Sylow 3-subgroup S and a Sylow 5-subgroup T of S7.

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Consider The Symmetric Group Sy Of Degree 7 I Give An Example Of A Sylow 3 Subgroup S And A Sylow 5 Subgroup T Of S7 1 (94.38 KiB) Viewed 33 times

Consider the symmetric group Sy of degree 7. (i) Give an example of a Sylow 3-subgroup S and a Sylow 5-subgroup T of S7. (ii) Determine the number of elements of order 5 in St. Hence, or other- wise, determine the number of Sylow 5-subgroups of St. (iii) Determine the order of the normaliser Ns,(T) of T in S7. (iv) Find an element o of order 4 in Sy that lies in Ns,(T), and an element T of order 2 in Sy that commutes with both T and o. (v) Prove that (T,0,T) = NS(T). =