# Aut(H) ≈ Inn(H). Show that G ≈ H x C(H) if H is normal in G.

• May 6th 2013, 08:45 PM
jamiebog
abstract algebra
Abstract algebra help
• May 7th 2013, 03:43 PM
johng
Re: Aut(H) ≈ Inn(H). Show that G ≈ H x C(H) if H is normal in G.
Hi,
I assume you mean Inn(H) is the group of inner automorphisms of H and when you say Aut(H) $\cong$Inn(H), you really mean Aut(H)=Inn(H) -- the inner automorphisms are automorphisms. Also from your conclusion, you left out the necessary assumption that Z(H) is trivial.

Assume H is a normal subgroup of G, Z(H) is trivial and Aut(H)=Inn(H). To show $G\cong H\times C_G(H)$, it is sufficient to show H and CG(H) are normal, $H\cap C_G(H)=<1>$ and $G=HC_G(H)$. Since the centralizer is a normal subgroup and Z(H)=<1>, the only thing to prove is $G=HC_G(H)$:

Let $x\in G$. Then conjugation of elements of H by x is an automorphism of H. So there exists $y\in H$ with $x^{-1}hx=y^{-1}hy$ for all $h\in H$. That is, $h=xy^{-1}hyx^{-1}$ for all h in H and so $xy{-1}\in C_G(H)$. Thus $x\in C_G(H)H=HC_G(H)$.