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).
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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![5. Let \( G \) be a group and \( H \leq G \).
(a) Prove for any \( a \in G \), \( aHa^{-1} \leq G \).
(b) Let \( x \in G \) and
\[ C(x) = \{ a \in G : ax = xa \} . \]
Prove \( C(x) \leq G \).
(c) Let
\[ N_G(H) = \{ a \in G : aH = Ha \} . \]
Show that \( N_G(H) \leq G \) and \( H \leq N_G(H) \).
(d) Let
\[ N = \bigcap_{x \in G} xHx^{-1} . \]
Show that \( N \leq G \) and for any \( a \in G \), \( aNa^{-1} = N \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97e612ef-1556-436b-b62c-352b280e9e69%2Fc4428d07-a449-48d3-8a2e-58fabdec935a%2F53v9mf_processed.png&w=3840&q=75)
Transcribed Image Text:5. Let \( G \) be a group and \( H \leq G \).
(a) Prove for any \( a \in G \), \( aHa^{-1} \leq G \).
(b) Let \( x \in G \) and
\[ C(x) = \{ a \in G : ax = xa \} . \]
Prove \( C(x) \leq G \).
(c) Let
\[ N_G(H) = \{ a \in G : aH = Ha \} . \]
Show that \( N_G(H) \leq G \) and \( H \leq N_G(H) \).
(d) Let
\[ N = \bigcap_{x \in G} xHx^{-1} . \]
Show that \( N \leq G \) and for any \( a \in G \), \( aNa^{-1} = N \).
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Vital instruction : According to Bartleby Guideline , i can solve at most first three subparts if student doesn't mention how many questions are to be solved . So , i will solve only first three subparts .
Subgroup definition : A non-empty subset H of G is a subgroup of G iff
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