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Let G be a group and H ≤ G. (a) Prove for any a € G, aHa-¹ ≤G. (b) Let x E G and Prove C (x) ≤ G. (c) Let C(x) = {a e G: ax = = xa}. NG (H) = {a e G: aH = Ha}. Show that NG (H) ≤ G and H ≤ NG (H).

Let G be a group and H ≤ G. (a) Prove for any a € G, aHa-¹ ≤G. (b) Let x E G and Prove C (x) ≤ G. (c) Let C(x) = {a e G: ax = = xa}. NG (H) = {a e G: aH = Ha}. Show that NG (H) ≤ G and H ≤ NG (H).

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.8: Some Results On Finite Abelian Groups (optional)
Problem 7E: Let G be a group and gG. Prove that if H is a Sylow p-group of G, then so is gHg1
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5. Let G be a group and H ≤ G. (a) Prove for any a € G, aHa-¹ <G. (b) Let x E G and Prove C (x) ≤ G. (c) Let C(x) = {a € G: ax = xa}. (d) Let NG (H) = {a e G : aH = Ha}. Show that NG (H) ≤ G and H ≤ NG (H). N = || xHx¯¹. xEG Show that N≤G and for any a € G, aNa-¹ = N.
Transcribed Image Text:5. Let G be a group and H ≤ G. (a) Prove for any a € G, aHa-¹ <G. (b) Let x E G and Prove C (x) ≤ G. (c) Let C(x) = {a € G: ax = xa}. (d) Let NG (H) = {a e G : aH = Ha}. Show that NG (H) ≤ G and H ≤ NG (H). N = || xHx¯¹. xEG Show that N≤G and for any a € G, aNa-¹ = N.
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