Correct formulation of problem $1.D.19$ from Isaacs's Finite Group Theory book
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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 $G/(Fcap C)$ instead of $G/(Gcap F)$ 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 $G/(Fcap C)$ 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?
group-theory finite-groups
edited Dec 21 '18 at 10:19
Nicky Hekster
28.6k63456
asked Dec 21 '18 at 7:09
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
$endgroup$
add a comment |
$begingroup$
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 $G/(Fcap C)$ instead of $G/(Gcap F)$ 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 $G/(Fcap C)$ 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?
group-theory finite-groups
edited Dec 21 '18 at 10:19
Nicky Hekster
28.6k63456
asked Dec 21 '18 at 7:09
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
$endgroup$
5
$begingroup$
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 $C/(C cap F)$.
$endgroup$
– Derek Holt
Dec 21 '18 at 8:34
$begingroup$
@DerekHolt, thank you very much!
$endgroup$
– Mikhail Goltvanitsa
Dec 21 '18 at 8:36
add a comment |
$begingroup$
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 $G/(Fcap C)$ instead of $G/(Gcap F)$ 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 $G/(Fcap C)$ 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?
group-theory finite-groups
edited Dec 21 '18 at 10:19
Nicky Hekster
28.6k63456
asked Dec 21 '18 at 7:09
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
$endgroup$
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 $G/(Fcap C)$ instead of $G/(Gcap F)$ 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 $G/(Fcap C)$ 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?
group-theory finite-groups
group-theory finite-groups
edited Dec 21 '18 at 10:19
Nicky Hekster
28.6k63456
asked Dec 21 '18 at 7:09
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
edited Dec 21 '18 at 10:19
Nicky Hekster
28.6k63456
asked Dec 21 '18 at 7:09
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
edited Dec 21 '18 at 10:19
Nicky Hekster
28.6k63456
edited Dec 21 '18 at 10:19
Nicky Hekster
28.6k63456
edited Dec 21 '18 at 10:19
Nicky Hekster
28.6k63456
28.6k63456
asked Dec 21 '18 at 7:09
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
asked Dec 21 '18 at 7:09
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
asked Dec 21 '18 at 7:09
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
623414
5
$begingroup$
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 $C/(C cap F)$.
$endgroup$
– Derek Holt
Dec 21 '18 at 8:34
$begingroup$
@DerekHolt, thank you very much!
$endgroup$
– Mikhail Goltvanitsa
Dec 21 '18 at 8:36
add a comment |
5
$begingroup$
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 $C/(C cap F)$.
$endgroup$
– Derek Holt
Dec 21 '18 at 8:34
$begingroup$
@DerekHolt, thank you very much!
$endgroup$
– Mikhail Goltvanitsa
Dec 21 '18 at 8:36
5
5
$begingroup$
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 $C/(C cap F)$.
$endgroup$
– Derek Holt
Dec 21 '18 at 8:34
$begingroup$
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 $C/(C cap F)$.
$endgroup$
– Derek Holt
Dec 21 '18 at 8:34
$begingroup$
@DerekHolt, thank you very much!
$endgroup$
– Mikhail Goltvanitsa
Dec 21 '18 at 8:36
$begingroup$
@DerekHolt, thank you very much!
$endgroup$
– Mikhail Goltvanitsa
Dec 21 '18 at 8:36
add a comment |
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5
$begingroup$
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 $C/(C cap F)$.
$endgroup$
– Derek Holt
Dec 21 '18 at 8:34
$begingroup$
@DerekHolt, thank you very much!
$endgroup$
– Mikhail Goltvanitsa
Dec 21 '18 at 8:36