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Let G be a finite group, let P E Syl, (G), and let P act on Syl,(G) by conjugation (see the Outline of Proof for Theorem 7.16). Assume that Q E Syl, (G) is fixed by this action, and let N P< N. Using Problem 7.3.5, conclude that Q = P. NG(Q). Show that d. %3D

Let G be a finite group, let P E Syl, (G), and let P act on Syl,(G) by conjugation (see the Outline of Proof for Theorem 7.16). Assume that Q E Syl, (G) is fixed by this action, and let N P< N. Using Problem 7.3.5, conclude that Q = P. NG(Q). Show that d. %3D

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 5E
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7.3.5. Let G be a finite group, P E Syl, (G), and N = NG(P). (a) Show that P e Syl,(N). -.3. The Number and Conjugacy of Sylow Subgroups* 153 (b) Show that the normalizer in N of P is N. In other words, NN(P) = N. (c) Show that P is the unique Sylow p-subgroup of N. In other words, |Syl,(N)| = 1.
Transcribed Image Text:7.3.5. Let G be a finite group, P E Syl, (G), and N = NG(P). (a) Show that P e Syl,(N). -.3. The Number and Conjugacy of Sylow Subgroups* 153 (b) Show that the normalizer in N of P is N. In other words, NN(P) = N. (c) Show that P is the unique Sylow p-subgroup of N. In other words, |Syl,(N)| = 1.
Let G be a finite group, let P E Syl,(G), and let P act on Syl,(G) by conjugation (see the Outline of Proof for Theorem 7.16). Assume that Q E Syl, (G) is fixed by this action, and let N = NG(Q). Show that P< N. Using Problem 7.3.5, conclude that Q = P.
Transcribed Image Text:Let G be a finite group, let P E Syl,(G), and let P act on Syl,(G) by conjugation (see the Outline of Proof for Theorem 7.16). Assume that Q E Syl, (G) is fixed by this action, and let N = NG(Q). Show that P< N. Using Problem 7.3.5, conclude that Q = P.
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