Let $H$ be a subgroup of $G$. Show that the centralizer of $H$ in $G$ is a normal subgroup of $N_G(H)$. Show also that the homomorphism $c : G \to \operatorname{Aut}(G)$ given by conjugation induces an injective homomorphism $N_G(H)/C_G(H) \to \operatorname{Aut}(H)$.

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    $\begingroup$ Perhaps post one question at a time and show the work you have attempted. Then you are much more likely to have people give you feedback. $\endgroup$ – Michael Joyce Apr 18 '13 at 2:10
  • $\begingroup$ The answer's hidden in the second proposition, though the description is slightly chaotic... $\endgroup$ – DonAntonio Apr 18 '13 at 2:14

I think you mean the map

$$\phi : G\to\operatorname{Aut}(G)\;,\;\;\phi(x):=I_x$$

where we define

$$I_x(g)=xgx^{-1}\;,\;\;\forall\,g\in G$$

(1) Show the above map is a homomorphism

(2) Find the above map's kernel

(3) Apply the first isomorphism theorem, taking into account that $\,\phi(G):=\operatorname{Inn}(G)\le\operatorname{Aut}(G)\,$


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