I need help with number 3 (f) Use Sylow's Theorem to show why every 8-cycle is contained in some 2-Sylow subgroup,and why every 2-Sylow subgroup of S8 has to contain an 8-cycle.(g) Find an 8-cycle in the 2-Sylow subgroup I described for you! Explain your answer.3. Let G be a finite group, and suppose that N is a normal subgroup of G. Suppose that P is ap-Sylow subgroup of N. (This will not necessarily be a p-Sylow subgroup of G.) Suppose that P isnormal in N. Prove that P is normal in G. (This should be short ... just be clear in your proof.)

Given that G be a finite group. Suppose N is normal subgroup of G. Also given that P is a p-Sylow subgroup of N. We know that a subgroup N is said to normal subgroup of group G, if for all g∈G, we have gNg-1=N. We know that, if order of subgroup P is m, then for all g∈G, we have OP=OgPg-1. Let g∈G is an arbitrary element. Given that N is normal subgroup of G, therefore for all g∈G, we have gNg-1=N. Let g∈G be an arbitrary element. Since P is a subgroup of N, therefore: gPg-1 is a subgroup of gNg-1. Since gNg-1=N, therefore gPg-1 is subgroup of N.Since OP=OgPg-1, therefore gPg-1 is a p-Sylow subgroup of N. We know that, by second Sylow theorem that any two p-Sylow subgroup are conjugate to each other. Since P and gPg-1 are two p-Sylow subgroups of N, therefore there exists h∈N such that: hPh-1=gPg-1. Since P is normal subgroup of N, therefore hPh-1=P for all h∈N. Therefore: hPh-1=gPg-1P=gPg-1 Therefore P=gPg-1. Since g∈G was an arbitrary element and P=gPg-1, therefore P is normal subgroup of G.