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…

3. Let G be a finite group, and suppose that N is a normal subgroup of G. Suppose that P is a p-Sylow subgroup of N. (This will not necessarily be a p-Sylow subgroup of G.) Suppose that P is normal in N. Prove that P is normal in G. (This should be short... just be clear in your proof.)

3. Let G be a finite group, and suppose that N is a normal subgroup of G. Suppose that P is a p-Sylow subgroup of N. (This will not necessarily be a p-Sylow subgroup of G.) Suppose that P is normal in N. Prove that P is normal in G. (This should be short... just be clear in your proof.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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 a
p-Sylow subgroup of N. (This will not necessarily be a p-Sylow subgroup of G.) Suppose that P is
normal in N. Prove that P is normal in G. (This should be short ... just be clear in your proof.)

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