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I think that there is a mispriprint in the formulation of problem $1.D.19$ from Isaacs's book Finite Group Theory (FGT). I give it here

Let $F = F(G)$, where $G$ is an arbitrary finite group, and let $C = C_G(F)$. Show that $G/(G \cap F)$ has no non-trivial abelian normal subgroup.

Here $F(G)$ is a Fitting subgroup of $G$. I think that there should be $G/(F\cap C)$ instead of $G/(G\cap F)$ since $G \cap F = F$ and in this case the result doesn't depend on $C$. On the other hand if the correct formulation is $G/(F \cap C)$, then the result is not valid for example for $G =D_8$. Since in this case $G=F$, $C = Z(G)$ and $G/(F\cap C)$ is isomorpic to $C_2 \times C_2$, wich of course has nontrivial abelian normal subgroup.

So, what is the correct formulation of this problem?

Martin Brandenburg
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Mikhail Goltvanitsa
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    It should be $C/(C \cap F)$. You could either prove it directly, or deduce it from the result that $C \le F$ when $G$ is solvable. I must have a kore recent edition of the book, because mine says $C/(C \cap F)$. – Derek Holt Dec 21 '18 at 08:34
  • Be also aware that the hint should read $\mathbf F(C)\triangleleft G$ and not $\mathbf F(G)\triangleleft G$ . – Leandro Caniglia Feb 25 '23 at 13:50
  • See also https://math.stackexchange.com/q/4469662/269050 – Leandro Caniglia Feb 26 '23 at 09:47

1 Answers1

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As it is mentioned in the comments the correct formulation is $C/(C \cap F)$ has no nontrivial abelian normal subgroup. To prove this you may use the following hints:

  1. Let $G$ be solvable then $G$ is non trivial if and only if $\mathbf{F}(G)$ is non trivial.

  2. If $G$ is a finite group then $\mathbf{F}(G/\mathbf{Z}(G))= \mathbf{F}(G)/\mathbf{Z}(G).$

  3. Show that $\mathbf{Z}(\mathbf{C}_G(\mathbf{F}(G)))= \mathbf{F}(\mathbf{C}_G(\mathbf{F}(G)))$ and apply Hint 2 in $\mathbf{C}_G(\mathbf{F}(G))$ and conclude that $\mathbf{F}(C/C \cap F)$ is trivial.

Q.E.D
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