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I am working on a problem on automorphism group of radical of finite group like this one:

##Aut(R(G))## is an automorphism group, whose elements consist of isomorphic mappings from ##R(G)## to itself. For visualization purpose, I envision the following example, where ##g \in G, r \in R(G)##:

I think these are too long to be useful in solving this problem, what I am looking for is a working definition, or the salient property of ##R(G)## to solve this problem. I would therefore appreciate any help or hints in solving this problem. Thanks for your time and help.

Here are what I know and what I don't know:Assume that ##R(G)## is simple and not commutative, show that ##G## is a subgroup of ##Aut(R(G)).##

##Aut(R(G))## is an automorphism group, whose elements consist of isomorphic mappings from ##R(G)## to itself. For visualization purpose, I envision the following example, where ##g \in G, r \in R(G)##:

## \begin{align} (\phi_g) \in Aut(R(G)), \quad \phi_g &: R(G) \to R(G), \\

&: r \mapsto r^g \\

&: r \mapsto grg^{-1} \end{align}##

Please correct me if I was wrong. But after this, I am stuck on how to prove that ##G## is subgroup of ##Aut(R(G)),## for to me it is like relating apple to orange. Perhaps this is because I failed to understand what the radical of finite group ##R(G)## stands for. For your info, in the class note the definition of ##R(G)## is a long and winding chain that goes like these:&: r \mapsto r^g \\

&: r \mapsto grg^{-1} \end{align}##

(a) ##R(G) := E(G)F(G); ##

(b) Where ##F(G)## is defined to be the (complex) product of all subgroups ##O_p(G)## with ##p## a prime number, this group ##F(G)## is called the

(c) And ##E(G)## is defined to be the subgroup of ##G## generated by all

(b) Where ##F(G)## is defined to be the (complex) product of all subgroups ##O_p(G)## with ##p## a prime number, this group ##F(G)## is called the

*Fitting*subgroup of ##G;##(c) And ##E(G)## is defined to be the subgroup of ##G## generated by all

*components*of ##G,## this group ##E(G)## is called the*Layer*of ##G##. And then another chain of definitions: A subnormal subgroup of ##G## is called a*component*of ##G## if it is*quasisimple*; the group ##G## is called*quasisimple*if ##G′ = G## and ##G/Z(G)## is simple.I think these are too long to be useful in solving this problem, what I am looking for is a working definition, or the salient property of ##R(G)## to solve this problem. I would therefore appreciate any help or hints in solving this problem. Thanks for your time and help.

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