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Let $G$ be a finite group: If $N$ is a minimal normal subgroup of $G$, then $F(G) \leq C_G(N)$.

Here $C_G(N)$ denotes the centralizer of $N$ in $G$, and $F(G)$ denotes the Fitting subgroup of $G$.

Martin Brandenburg
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M.Mazoo
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1 Answers1

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If $N$ is abelian, then $N \leq F$ and $[F,N] < N$ is still normal, so $[F,N]=1$ as claimed. If $N$ is non-abelian, then $[F,N] \leq F \cap N = 1$ as claimed.

You may try to prove sort of a converse: $F(G)$ is the intersection of the centralizers, not just of the minimal normal subgroups, but of all $K/L$ where $K$ is a minimal normal subgroup of $G/L$.

Jack Schmidt
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  • Why [F,N]≤F∩N=1........ F intersection of N =1 (WHY)? – M.Mazoo Jun 07 '13 at 20:02
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    If $N$ is nonabelian, then it is not nilpotent, so $N \not\leq F$. So, $F \cap N < N$ and $N$ is minimal normal. Since $F \cap N$ is normal too, the only possibility is $F \cap N = 1$. – Jack Schmidt Jun 07 '13 at 20:18
  • Are u sure "
    If N is nonabelian, then it is not nilpotent" ?     – M.Mazoo Jun 07 '13 at 20:29
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    Yes, because $N$ is a minimal normal subgroup, but $[N,N]$ is another normal subgroup. If it is $1$, then $N$ is abelian. If not, then $[N,N]=N$ and $N$ is not nilpotent. – Jack Schmidt Jun 07 '13 at 20:35
  • Tank you. parden me "if i have question can i mail you ?" – M.Mazoo Jun 07 '13 at 20:39
  • This website is better. During the school year I have 600 students, so it is difficult to keep track of emails. – Jack Schmidt Jun 07 '13 at 20:42
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    Hi, Parden me i have just one question if N is abelian, [F,N]=N not can true Why ? – M.Mazoo Jun 08 '13 at 09:53
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    If N is abelian, then N ≤ F. F is a nilpotent group and nilpotent groups have a nice property: if X is a non-identity normal subgroup of F, then [F,X] < X. This is called “nilpotent action”. Proof: Assume by way of contradiction that [F,X] = X. Then [F,[F,X]] = [F,X] = X, so X = [F,[F,....,[F,X]...]] ≤ [F,[F,....,[F,F]..]] = 1 (if there are enough [F, ...). – Jack Schmidt Jun 08 '13 at 18:57
  • Completly i got it. tank you tank you. I wish you could see it up close. – M.Mazoo Jun 08 '13 at 19:08