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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.)

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.5: Normal Subgroups
Problem 14E: 14. Find groups and such that and the following conditions are satisfied: a. is a normal...
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(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.)
Transcribed Image Text:(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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