Let $G$ be a group and $H$ be any subgroup. It is known that for a subgroup $H$, $C(H)$ is a normal subgroup of $N(H)$. So we can talk about its quotient groups. Is there any other isomorphism from the quotient group $N(H)/C(H)$ to any other subgroup of $G$ other than the natural isomorphism $N(H)/C(H)$ is isomorphic to $N(H)$.
[Math] Quotient group formed by normalizer and centralizer of a subgroup $H$
abstract-algebragroup-theory
Best Answer
The normalizer $N_G(H)$ acts on $H$ by conjugation and the kernel of this action is exactly $C_G(H)$. Hence $N_G(H)/C_G(H)$ can be homomorphically embedded in Aut$(H)$. An small application: if $H$ is normal and $|H|=p$, the smallest prime dividing $|G|$, then $H \subseteq Z(G)$.