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
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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 \( a \in G \), \( aNa^{-1} = N \).
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 a ,bH ab-1H

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