- #1

- 85

- 0

I am working on this problem with lots and lots of nesting definitions like this following, and I have been trying to get help from

For your info, ##R(G)## is called the

And then the problem comes with a hint: Use Exercise 3.4. This exercise has the following mappings:

therefore I think that this hint is directing me to use conjugate automorphism.

I was thinking about letting ##\varphi : G/Z(R(G)) \to Aut(R(G))##, and thenstructuring this ##\varphi## into mappings like (1) and (2), so that at least I have a visual idea. But after getting limited responses from my earlier postings, I am not so sure that will be the right direction. Any help or hints would be very, very much appreciated.

Thank you for your time and help.

*here*as well as http://www.quora.com/How-do-I-prove-G-Z-R-G-is-isomorphic-to-Aut-R-G [Broken], but none gave me complete help:Show that ##G/Z(R(G))## is isomorphic to a subgroup of ##Aut(R(G))##.

For your info, ##R(G)## is called the

*Radical*of ##G##, defined as ##R(G) := E(G)F(G)##, where ##E(G)## is called the*Layer*of ##G##, and ##F(G)## is called the*Fitting*of ##G##. Long story short, it suffices (I think) to say that since both ##E(G)## and ##F(G)## are normal according to a lemma, therefore ##R(G)## is normal too.And then the problem comes with a hint: Use Exercise 3.4. This exercise has the following mappings:

##\begin{align}

\alpha \ &: \ N_G(H) \to Aut(H), \quad g \mapsto \alpha_g \tag{1}\\

\text{where} \ \alpha_g \ &: \ H \to H, \quad h \mapsto h^g, \tag{2}

\end{align}##

\alpha \ &: \ N_G(H) \to Aut(H), \quad g \mapsto \alpha_g \tag{1}\\

\text{where} \ \alpha_g \ &: \ H \to H, \quad h \mapsto h^g, \tag{2}

\end{align}##

therefore I think that this hint is directing me to use conjugate automorphism.

I was thinking about letting ##\varphi : G/Z(R(G)) \to Aut(R(G))##, and thenstructuring this ##\varphi## into mappings like (1) and (2), so that at least I have a visual idea. But after getting limited responses from my earlier postings, I am not so sure that will be the right direction. Any help or hints would be very, very much appreciated.

Thank you for your time and help.

Last edited by a moderator: