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Math Help - normal subgroup

  1. #1
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    normal subgroup

    Let G be a finite group and H a subgroup of G. Let N(H) = {a∈G: axaˉ∈ H for every x∈H}. Prove a∈N(H) implies that aH aˉ∈H. Prove also that N(H) is a subgroup of G. Prove that H⊆N(H); indeed, H is normal subgroup of N(H)
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  2. #2
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    Quote Originally Posted by apple2009 View Post
    Let G be a finite group and H a subgroup of G. Let N(H) = {a∈G: axaˉ∈ H for every x∈H}. Prove a∈N(H) implies that aH aˉ∈H.

    This is the exact definition of N_G(H) you wrote three words before. There's nothing to prove here.

    Prove also that N(H) is a subgroup of G. Prove that H⊆N(H); indeed, H is normal subgroup of N(H)

    This is just applying both the definition of normalizer and of normal subgroup. If you have doubts re-read the definitions and understand them.

    Tonio
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