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 $N(H)$.

  • $\begingroup$ There is no natural isomorphism between $N(H)/C(H)$ and $N(H)$ ... $\endgroup$ – Nicky Hekster Nov 21 '16 at 20:26

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 $H \subseteq Z(G)$.

  • $\begingroup$ Thank u proffesor. Could you list some books of advanced group theory? $\endgroup$ – Shiksharthi Sharma Nov 23 '16 at 2:28
  • $\begingroup$ Yes certainly. An excellent book is that of I.M. Isaacs, Finite Group Theory. Another one, somewhat older but very well written and with an abundance of exercises is J.S. Rose, A Course on Group Theory. You will find both books on Google Books, so that you can see its contents. If you want to study infinite groups, there is also a wealth of books. You might want to start with D.J.S. Robinson, A Course in the Theory of Groups. I can also recommend math.uconn.edu/~kconrad/blurbs $\endgroup$ – Nicky Hekster Nov 23 '16 at 10:24

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