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3. (a) It follows from Sylow's (First) Theorem that if G is a finite p-group (for some prime p), that any subgroup H with index p is normal in G. Show this directly using the action of H on G/H (the left cosets of H) given by left multiplication. (b) Let G be a group with order p"m (p a prime, p does not divide m). Let P be the intersection of all Sylow p-subgroups of G. Prove that P is normal in G. Show further that any other normal p-subgroup of G is contained in P. (c) Let p be a prime. Find all groups up to isomorphism of order 2p. Justify your answer. Hint: The answer is 2. Also you can do this without Sylow's theorems, but they provide a lot of shortcuts to your argument.

3. (a) It follows from Sylow's (First) Theorem that if G is a finite p-group (for some prime p), that any subgroup H with index p is normal in G. Show this directly using the action of H on G/H (the left cosets of H) given by left multiplication. (b) Let G be a group with order p"m (p a prime, p does not divide m). Let P be the intersection of all Sylow p-subgroups of G. Prove that P is normal in G. Show further that any other normal p-subgroup of G is contained in P. (c) Let p be a prime. Find all groups up to isomorphism of order 2p. Justify your answer. Hint: The answer is 2. Also you can do this without Sylow's theorems, but they provide a lot of shortcuts to your argument.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.5: Isomorphisms
Problem 8E: Exercises 8. Find an isomorphism from the group in Example of this section to the multiplicative...
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3. (a) It follows from Sylow's (First) Theorem that if G is a finite p-group (for some prime p), that any subgroup H with index p is normal in G. Show this directly using the action of H on G/H (the left cosets of H) given by left multiplication. (b) Let G be a group with order pm (p a prime, p does not divide m). Let P be the intersection of all Sylow p-subgroups of G. Prove that P is normal in G. Show further that any other normal p-subgroup of G is contained in P. (c) Let p be a prime. Find all groups up to isomorphism of order 2p. Justify your answer. Hint: The answer is 2. Also you can do this without Sylow's theorems, but they provide a lot of shortcuts to your argument.
Transcribed Image Text:3. (a) It follows from Sylow's (First) Theorem that if G is a finite p-group (for some prime p), that any subgroup H with index p is normal in G. Show this directly using the action of H on G/H (the left cosets of H) given by left multiplication. (b) Let G be a group with order pm (p a prime, p does not divide m). Let P be the intersection of all Sylow p-subgroups of G. Prove that P is normal in G. Show further that any other normal p-subgroup of G is contained in P. (c) Let p be a prime. Find all groups up to isomorphism of order 2p. Justify your answer. Hint: The answer is 2. Also you can do this without Sylow's theorems, but they provide a lot of shortcuts to your argument.
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