Subgroups of direct products of free groups
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I am reading the following paper of Miller: http://researchers.ms.unimelb.edu.au/
He says that if $G= F_{1} times F_{2}$ is a direct product of two free groups and $H$ is a subgroup of $G$, then it can be assumed that the projection maps $$text{p}_{i} colon H rightarrow F_{i}$$ are surjective.
I do not understand why is this trivial. If ${f_{1},cdots,f_{n}}$ is a basis of the free group $F_{1}$, why should we have elements of the form $(f_{i},h_{i})$ in $H$ for all $iin {1,cdots,n}$? He simply says that this follows because subgroups of free groups are free.
group-theory free-groups direct-product
asked Jan 3 at 17:33
KarenKaren
946
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add a comment |
$begingroup$
I am reading the following paper of Miller: http://researchers.ms.unimelb.edu.au/
He says that if $G= F_{1} times F_{2}$ is a direct product of two free groups and $H$ is a subgroup of $G$, then it can be assumed that the projection maps $$text{p}_{i} colon H rightarrow F_{i}$$ are surjective.
I do not understand why is this trivial. If ${f_{1},cdots,f_{n}}$ is a basis of the free group $F_{1}$, why should we have elements of the form $(f_{i},h_{i})$ in $H$ for all $iin {1,cdots,n}$? He simply says that this follows because subgroups of free groups are free.
group-theory free-groups direct-product
asked Jan 3 at 17:33
KarenKaren
946
$endgroup$
3
$begingroup$
Let $H_isubset F_i$ be the image of $H$ under $p_i$. Then $H_i$ is free and $Hsubset H_1times H_2$
$endgroup$
– Max
Jan 3 at 17:37
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Omg it was trivial and I have been ages thinking about it! Thank you very much @Max :)
$endgroup$
– Karen
Jan 3 at 17:44
1
$begingroup$
Don't beat yourself up about it, @Karen; it took over three hundred pages of dense symbol shuffling to get to a proof that, in fact, yes, $1+1=2$ in Russell & Whitehead's "Principia Mathematica".
$endgroup$
– Shaun
Jan 3 at 17:51
add a comment |
$begingroup$
I am reading the following paper of Miller: http://researchers.ms.unimelb.edu.au/
He says that if $G= F_{1} times F_{2}$ is a direct product of two free groups and $H$ is a subgroup of $G$, then it can be assumed that the projection maps $$text{p}_{i} colon H rightarrow F_{i}$$ are surjective.
I do not understand why is this trivial. If ${f_{1},cdots,f_{n}}$ is a basis of the free group $F_{1}$, why should we have elements of the form $(f_{i},h_{i})$ in $H$ for all $iin {1,cdots,n}$? He simply says that this follows because subgroups of free groups are free.
group-theory free-groups direct-product
asked Jan 3 at 17:33
KarenKaren
946
$endgroup$
I am reading the following paper of Miller: http://researchers.ms.unimelb.edu.au/
He says that if $G= F_{1} times F_{2}$ is a direct product of two free groups and $H$ is a subgroup of $G$, then it can be assumed that the projection maps $$text{p}_{i} colon H rightarrow F_{i}$$ are surjective.
I do not understand why is this trivial. If ${f_{1},cdots,f_{n}}$ is a basis of the free group $F_{1}$, why should we have elements of the form $(f_{i},h_{i})$ in $H$ for all $iin {1,cdots,n}$? He simply says that this follows because subgroups of free groups are free.
group-theory free-groups direct-product
group-theory free-groups direct-product
asked Jan 3 at 17:33
KarenKaren
946
asked Jan 3 at 17:33
KarenKaren
946
asked Jan 3 at 17:33
KarenKaren
946
asked Jan 3 at 17:33
KarenKaren
946
asked Jan 3 at 17:33
KarenKaren
946
946
3
$begingroup$
Let $H_isubset F_i$ be the image of $H$ under $p_i$. Then $H_i$ is free and $Hsubset H_1times H_2$
$endgroup$
– Max
Jan 3 at 17:37
$begingroup$
Omg it was trivial and I have been ages thinking about it! Thank you very much @Max :)
$endgroup$
– Karen
Jan 3 at 17:44
1
$begingroup$
Don't beat yourself up about it, @Karen; it took over three hundred pages of dense symbol shuffling to get to a proof that, in fact, yes, $1+1=2$ in Russell & Whitehead's "Principia Mathematica".
$endgroup$
– Shaun
Jan 3 at 17:51
add a comment |
3
$begingroup$
Let $H_isubset F_i$ be the image of $H$ under $p_i$. Then $H_i$ is free and $Hsubset H_1times H_2$
$endgroup$
– Max
Jan 3 at 17:37
$begingroup$
Omg it was trivial and I have been ages thinking about it! Thank you very much @Max :)
$endgroup$
– Karen
Jan 3 at 17:44
1
$begingroup$
Don't beat yourself up about it, @Karen; it took over three hundred pages of dense symbol shuffling to get to a proof that, in fact, yes, $1+1=2$ in Russell & Whitehead's "Principia Mathematica".
$endgroup$
– Shaun
Jan 3 at 17:51
3
3
$begingroup$
Let $H_isubset F_i$ be the image of $H$ under $p_i$. Then $H_i$ is free and $Hsubset H_1times H_2$
$endgroup$
– Max
Jan 3 at 17:37
$begingroup$
Let $H_isubset F_i$ be the image of $H$ under $p_i$. Then $H_i$ is free and $Hsubset H_1times H_2$
$endgroup$
– Max
Jan 3 at 17:37
$begingroup$
Omg it was trivial and I have been ages thinking about it! Thank you very much @Max :)
$endgroup$
– Karen
Jan 3 at 17:44
$begingroup$
Omg it was trivial and I have been ages thinking about it! Thank you very much @Max :)
$endgroup$
– Karen
Jan 3 at 17:44
1
1
$begingroup$
Don't beat yourself up about it, @Karen; it took over three hundred pages of dense symbol shuffling to get to a proof that, in fact, yes, $1+1=2$ in Russell & Whitehead's "Principia Mathematica".
$endgroup$
– Shaun
Jan 3 at 17:51
$begingroup$
Don't beat yourself up about it, @Karen; it took over three hundred pages of dense symbol shuffling to get to a proof that, in fact, yes, $1+1=2$ in Russell & Whitehead's "Principia Mathematica".
$endgroup$
– Shaun
Jan 3 at 17:51
add a comment |
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3
$begingroup$
Let $H_isubset F_i$ be the image of $H$ under $p_i$. Then $H_i$ is free and $Hsubset H_1times H_2$
$endgroup$
– Max
Jan 3 at 17:37
$begingroup$
Omg it was trivial and I have been ages thinking about it! Thank you very much @Max :)
$endgroup$
– Karen
Jan 3 at 17:44
1
$begingroup$
Don't beat yourself up about it, @Karen; it took over three hundred pages of dense symbol shuffling to get to a proof that, in fact, yes, $1+1=2$ in Russell & Whitehead's "Principia Mathematica".
$endgroup$
– Shaun
Jan 3 at 17:51