Let H be a subgroup of group (G, o). (a) Show that the normalizer of H in G/cent

Question

Let H be a subgroup of group (G, o). (a) Show that the normalizer of H in G/centralizer of H in G is isomorphic to a subgroup aut(H). (b) Show that the centralizer of H in G = normalizer of H in G if lHl = 2 Show complete solutions in both

Explanation / Answer

normalizer NG(H) = {g in G|gHg(^-1)=H} centralizer CG(H) = {g in G|gh = hg for all h in H} 3. The attempt at a solution Okay, for part (1) I showed the left coset of H equals to the right coset of H, that is g*H = H*g = H h in H Can anyone show me an example that it is NOT necessarily true if H is NOT a subgroup? (If H is NOT a subgroup of G, isn't that obvious that it's not necessary a subgroup of its Normalizer?) For part (2) by definition of the centralizer CG(H)={g in G|gh = hg for all h in H}. (=>) Suppose H g = h1*g*h1^(-1) g*h2 = h2*g => g = h2*g*h2^(-1) So h1*g*h1^(-1) = h2*g*h2^(-1) Also h1*h2^(-1) is in H by definition of being subgroup of it's centralizer, so g*h1*h2^(-1)=h1*h2^(-1)*g Okay, I've been trying to drive to h1*h2=h2*h1 but somehow stuck. Can anyone help? I know it's not a complicated problem... ( h1*h2^(-1) = h2^(-1)*h1. WTS H h1*h2^(-1) is in H Obviously H is nonempty since the identity is there. Let h1, h2 in H Then since h1*h2=h2*h1 => h1*h2^(-1) = h2^(-1)*h1

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