Residually finite extension of a finite group
4
$begingroup$
Do you know how to prove that the extension of a finte group by a residually finite group is residually finite? I know that there is a result of Mal'cev proving something stronger, but I cannot find it either.
I say that a group $G$ is an extension of a finte group by a residually finite group, in this case, if there exists a finite normal subgroup $N$ of $G$ such that $G/N$ is residually finite.
group-theory normal-subgroups group-extensions
edited Dec 22 '18 at 15:02
Shaun
10.6k113687
asked Jan 2 '17 at 16:12
Alex DoeAlex Doe
135119
$endgroup$
|
show 3 more comments
4
$begingroup$
Do you know how to prove that the extension of a finte group by a residually finite group is residually finite? I know that there is a result of Mal'cev proving something stronger, but I cannot find it either.
I say that a group $G$ is an extension of a finte group by a residually finite group, in this case, if there exists a finite normal subgroup $N$ of $G$ such that $G/N$ is residually finite.
group-theory normal-subgroups group-extensions
edited Dec 22 '18 at 15:02
Shaun
10.6k113687
asked Jan 2 '17 at 16:12
Alex DoeAlex Doe
135119
$endgroup$
$begingroup$
Mal'cev proved in "On homomorphisms onto finite groups (1958)" that a split extension of a finitely generated residually finite group by a residually finite group is residually finite. Miller improved on this result in "On Group-Theoretic Decision Problems and their Classification (1971)"
$endgroup$
– TastyRomeo
Jan 2 '17 at 16:25
$begingroup$
What do you mean by extension of group A by group B? There are two possible meanings, and they are both used roughly equally often, so you cannot use this terminology without explaining which you mean.
$endgroup$
– Derek Holt
Jan 2 '17 at 16:25
1
$begingroup$
Just edited with my meaning for extension
$endgroup$
– Alex Doe
Jan 2 '17 at 16:31
$begingroup$
It may be helpful to do a few reductions. Since $C_G(N)$ has finite index in $G$, you can assume that $N le Z(G)$. You could then assume by induction on $|N|$ that $|N|$ is prime. I think the result is easy for abelian groups, and so you can assume that $N le [G,G]$, since otherwise it follows from looking at $G/[G,G]$. So $N$ is a quotient of the Schur Multiplier $H_2(G)$. That's as far as I have got!
$endgroup$
– Derek Holt
Jan 2 '17 at 16:43
2
$begingroup$
The result appears to be false. On math.umbc.edu/~campbell/CombGpThy/RF_Thesis/2_RF_Results.html in the section on extensions of residually finite groups, there is an example attributed to Kanta Gupta of a group $G$ that is not residually finite with a normal subgroup $K$ of order $2$ such that $Q cong G/K$ is residually finite.
$endgroup$
– Derek Holt
Jan 2 '17 at 17:22
|
show 3 more comments
4
4
4
2
$begingroup$
Do you know how to prove that the extension of a finte group by a residually finite group is residually finite? I know that there is a result of Mal'cev proving something stronger, but I cannot find it either.
I say that a group $G$ is an extension of a finte group by a residually finite group, in this case, if there exists a finite normal subgroup $N$ of $G$ such that $G/N$ is residually finite.
group-theory normal-subgroups group-extensions
edited Dec 22 '18 at 15:02
Shaun
10.6k113687
asked Jan 2 '17 at 16:12
Alex DoeAlex Doe
135119
$endgroup$
Do you know how to prove that the extension of a finte group by a residually finite group is residually finite? I know that there is a result of Mal'cev proving something stronger, but I cannot find it either.
I say that a group $G$ is an extension of a finte group by a residually finite group, in this case, if there exists a finite normal subgroup $N$ of $G$ such that $G/N$ is residually finite.
group-theory normal-subgroups group-extensions
group-theory normal-subgroups group-extensions
edited Dec 22 '18 at 15:02
Shaun
10.6k113687
asked Jan 2 '17 at 16:12
Alex DoeAlex Doe
135119
edited Dec 22 '18 at 15:02
Shaun
10.6k113687
asked Jan 2 '17 at 16:12
Alex DoeAlex Doe
135119
edited Dec 22 '18 at 15:02
Shaun
10.6k113687
edited Dec 22 '18 at 15:02
Shaun
10.6k113687
edited Dec 22 '18 at 15:02
Shaun
10.6k113687
10.6k113687
asked Jan 2 '17 at 16:12
Alex DoeAlex Doe
135119
asked Jan 2 '17 at 16:12
Alex DoeAlex Doe
135119
asked Jan 2 '17 at 16:12
Alex DoeAlex Doe
135119
135119
$begingroup$
Mal'cev proved in "On homomorphisms onto finite groups (1958)" that a split extension of a finitely generated residually finite group by a residually finite group is residually finite. Miller improved on this result in "On Group-Theoretic Decision Problems and their Classification (1971)"
$endgroup$
– TastyRomeo
Jan 2 '17 at 16:25
$begingroup$
What do you mean by extension of group A by group B? There are two possible meanings, and they are both used roughly equally often, so you cannot use this terminology without explaining which you mean.
$endgroup$
– Derek Holt
Jan 2 '17 at 16:25
1
$begingroup$
Just edited with my meaning for extension
$endgroup$
– Alex Doe
Jan 2 '17 at 16:31
$begingroup$
It may be helpful to do a few reductions. Since $C_G(N)$ has finite index in $G$, you can assume that $N le Z(G)$. You could then assume by induction on $|N|$ that $|N|$ is prime. I think the result is easy for abelian groups, and so you can assume that $N le [G,G]$, since otherwise it follows from looking at $G/[G,G]$. So $N$ is a quotient of the Schur Multiplier $H_2(G)$. That's as far as I have got!
$endgroup$
– Derek Holt
Jan 2 '17 at 16:43
2
$begingroup$
The result appears to be false. On math.umbc.edu/~campbell/CombGpThy/RF_Thesis/2_RF_Results.html in the section on extensions of residually finite groups, there is an example attributed to Kanta Gupta of a group $G$ that is not residually finite with a normal subgroup $K$ of order $2$ such that $Q cong G/K$ is residually finite.
$endgroup$
– Derek Holt
Jan 2 '17 at 17:22
|
show 3 more comments
$begingroup$
Mal'cev proved in "On homomorphisms onto finite groups (1958)" that a split extension of a finitely generated residually finite group by a residually finite group is residually finite. Miller improved on this result in "On Group-Theoretic Decision Problems and their Classification (1971)"
$endgroup$
– TastyRomeo
Jan 2 '17 at 16:25
$begingroup$
What do you mean by extension of group A by group B? There are two possible meanings, and they are both used roughly equally often, so you cannot use this terminology without explaining which you mean.
$endgroup$
– Derek Holt
Jan 2 '17 at 16:25
1
$begingroup$
Just edited with my meaning for extension
$endgroup$
– Alex Doe
Jan 2 '17 at 16:31
$begingroup$
It may be helpful to do a few reductions. Since $C_G(N)$ has finite index in $G$, you can assume that $N le Z(G)$. You could then assume by induction on $|N|$ that $|N|$ is prime. I think the result is easy for abelian groups, and so you can assume that $N le [G,G]$, since otherwise it follows from looking at $G/[G,G]$. So $N$ is a quotient of the Schur Multiplier $H_2(G)$. That's as far as I have got!
$endgroup$
– Derek Holt
Jan 2 '17 at 16:43
2
$begingroup$
The result appears to be false. On math.umbc.edu/~campbell/CombGpThy/RF_Thesis/2_RF_Results.html in the section on extensions of residually finite groups, there is an example attributed to Kanta Gupta of a group $G$ that is not residually finite with a normal subgroup $K$ of order $2$ such that $Q cong G/K$ is residually finite.
$endgroup$
– Derek Holt
Jan 2 '17 at 17:22
$begingroup$
Mal'cev proved in "On homomorphisms onto finite groups (1958)" that a split extension of a finitely generated residually finite group by a residually finite group is residually finite. Miller improved on this result in "On Group-Theoretic Decision Problems and their Classification (1971)"
$endgroup$
– TastyRomeo
Jan 2 '17 at 16:25
$begingroup$
Mal'cev proved in "On homomorphisms onto finite groups (1958)" that a split extension of a finitely generated residually finite group by a residually finite group is residually finite. Miller improved on this result in "On Group-Theoretic Decision Problems and their Classification (1971)"
$endgroup$
– TastyRomeo
Jan 2 '17 at 16:25
$begingroup$
What do you mean by extension of group A by group B? There are two possible meanings, and they are both used roughly equally often, so you cannot use this terminology without explaining which you mean.
$endgroup$
– Derek Holt
Jan 2 '17 at 16:25
$begingroup$
What do you mean by extension of group A by group B? There are two possible meanings, and they are both used roughly equally often, so you cannot use this terminology without explaining which you mean.
$endgroup$
– Derek Holt
Jan 2 '17 at 16:25
1
1
$begingroup$
Just edited with my meaning for extension
$endgroup$
– Alex Doe
Jan 2 '17 at 16:31
$begingroup$
Just edited with my meaning for extension
$endgroup$
– Alex Doe
Jan 2 '17 at 16:31
$begingroup$
It may be helpful to do a few reductions. Since $C_G(N)$ has finite index in $G$, you can assume that $N le Z(G)$. You could then assume by induction on $|N|$ that $|N|$ is prime. I think the result is easy for abelian groups, and so you can assume that $N le [G,G]$, since otherwise it follows from looking at $G/[G,G]$. So $N$ is a quotient of the Schur Multiplier $H_2(G)$. That's as far as I have got!
$endgroup$
– Derek Holt
Jan 2 '17 at 16:43
$begingroup$
It may be helpful to do a few reductions. Since $C_G(N)$ has finite index in $G$, you can assume that $N le Z(G)$. You could then assume by induction on $|N|$ that $|N|$ is prime. I think the result is easy for abelian groups, and so you can assume that $N le [G,G]$, since otherwise it follows from looking at $G/[G,G]$. So $N$ is a quotient of the Schur Multiplier $H_2(G)$. That's as far as I have got!
$endgroup$
– Derek Holt
Jan 2 '17 at 16:43
2
2
$begingroup$
The result appears to be false. On math.umbc.edu/~campbell/CombGpThy/RF_Thesis/2_RF_Results.html in the section on extensions of residually finite groups, there is an example attributed to Kanta Gupta of a group $G$ that is not residually finite with a normal subgroup $K$ of order $2$ such that $Q cong G/K$ is residually finite.
$endgroup$
– Derek Holt
Jan 2 '17 at 17:22
$begingroup$
The result appears to be false. On math.umbc.edu/~campbell/CombGpThy/RF_Thesis/2_RF_Results.html in the section on extensions of residually finite groups, there is an example attributed to Kanta Gupta of a group $G$ that is not residually finite with a normal subgroup $K$ of order $2$ such that $Q cong G/K$ is residually finite.
$endgroup$
– Derek Holt
Jan 2 '17 at 17:22
|
show 3 more comments
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$begingroup$
Mal'cev proved in "On homomorphisms onto finite groups (1958)" that a split extension of a finitely generated residually finite group by a residually finite group is residually finite. Miller improved on this result in "On Group-Theoretic Decision Problems and their Classification (1971)"
$endgroup$
– TastyRomeo
Jan 2 '17 at 16:25
$begingroup$
What do you mean by extension of group A by group B? There are two possible meanings, and they are both used roughly equally often, so you cannot use this terminology without explaining which you mean.
$endgroup$
– Derek Holt
Jan 2 '17 at 16:25
1
$begingroup$
Just edited with my meaning for extension
$endgroup$
– Alex Doe
Jan 2 '17 at 16:31
$begingroup$
It may be helpful to do a few reductions. Since $C_G(N)$ has finite index in $G$, you can assume that $N le Z(G)$. You could then assume by induction on $|N|$ that $|N|$ is prime. I think the result is easy for abelian groups, and so you can assume that $N le [G,G]$, since otherwise it follows from looking at $G/[G,G]$. So $N$ is a quotient of the Schur Multiplier $H_2(G)$. That's as far as I have got!
$endgroup$
– Derek Holt
Jan 2 '17 at 16:43
2
$begingroup$
The result appears to be false. On math.umbc.edu/~campbell/CombGpThy/RF_Thesis/2_RF_Results.html in the section on extensions of residually finite groups, there is an example attributed to Kanta Gupta of a group $G$ that is not residually finite with a normal subgroup $K$ of order $2$ such that $Q cong G/K$ is residually finite.
$endgroup$
– Derek Holt
Jan 2 '17 at 17:22