to be the subset Let H be a subgroup of G. Define the normalizer of H in NG(H) = {g e G| gHg H}. (i) Prove that NG(H) is a subgroup of G that contains H. (ii) Prove that H 4 NG(H). (iii) Prove that if H < K

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Let \( H \) be a subgroup of \( G \). Define the normalizer of \( H \) in \( G \) to be the subset

\[ N_G(H) = \{ g \in G \mid gHg^{-1} = H \}. \]

(i) Prove that \( N_G(H) \) is a subgroup of \( G \) that contains \( H \).

(ii) Prove that \( H \trianglelefteq N_G(H) \).

(iii) Prove that if \( H \leq K \leq G \), and \( H \trianglelefteq K \), then \( K \subseteq N_G(H) \).
Transcribed Image Text:Let \( H \) be a subgroup of \( G \). Define the normalizer of \( H \) in \( G \) to be the subset \[ N_G(H) = \{ g \in G \mid gHg^{-1} = H \}. \] (i) Prove that \( N_G(H) \) is a subgroup of \( G \) that contains \( H \). (ii) Prove that \( H \trianglelefteq N_G(H) \). (iii) Prove that if \( H \leq K \leq G \), and \( H \trianglelefteq K \), then \( K \subseteq N_G(H) \).
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