Closed subgroups of abelian groupsEmbedded Lie subgroups are closed.The injectivity of torus in the category of abelian Lie groupsCenter of compact lie group closed?Closedness of connected semisimple Lie subgroups of semisimple groupsIntersection of a family of closed Lie subgroupsExamples about maximal Abelian subgroup is not a maximal torus in compact connected Lie group $G$.A question on abelian Lie groups and maximal compact subgroupReal and complex nilpotent Lie groupsClosed Subgroups of Lie Groups are closed Lie Subgroups?Lattice and abelian Lie groups

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Closed subgroups of abelian groups


Embedded Lie subgroups are closed.The injectivity of torus in the category of abelian Lie groupsCenter of compact lie group closed?Closedness of connected semisimple Lie subgroups of semisimple groupsIntersection of a family of closed Lie subgroupsExamples about maximal Abelian subgroup is not a maximal torus in compact connected Lie group $G$.A question on abelian Lie groups and maximal compact subgroupReal and complex nilpotent Lie groupsClosed Subgroups of Lie Groups are closed Lie Subgroups?Lattice and abelian Lie groups













2












$begingroup$


What is an example of an abelian Lie group $G$ and a closed subgroup $H$ such that $Gnotcong G/H times H$?



Would the circle $S^1$ in $mathbb R^2$ be an example? what is $mathbb R^2/S^1$?










share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    Isn't $G=mathbbR$ and $H=mathbbZ$ an example of the non-iso you want? All you need to show is that $mathbbR$ is not iso to $S^1 times mathbbZ$. That's easy.
    $endgroup$
    – Randall
    2 hours ago
















2












$begingroup$


What is an example of an abelian Lie group $G$ and a closed subgroup $H$ such that $Gnotcong G/H times H$?



Would the circle $S^1$ in $mathbb R^2$ be an example? what is $mathbb R^2/S^1$?










share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    Isn't $G=mathbbR$ and $H=mathbbZ$ an example of the non-iso you want? All you need to show is that $mathbbR$ is not iso to $S^1 times mathbbZ$. That's easy.
    $endgroup$
    – Randall
    2 hours ago














2












2








2





$begingroup$


What is an example of an abelian Lie group $G$ and a closed subgroup $H$ such that $Gnotcong G/H times H$?



Would the circle $S^1$ in $mathbb R^2$ be an example? what is $mathbb R^2/S^1$?










share|cite|improve this question











$endgroup$




What is an example of an abelian Lie group $G$ and a closed subgroup $H$ such that $Gnotcong G/H times H$?



Would the circle $S^1$ in $mathbb R^2$ be an example? what is $mathbb R^2/S^1$?







general-topology differential-geometry lie-groups lie-algebras






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 4 hours ago









Clayton

19.6k33288




19.6k33288










asked 4 hours ago









Amrat AAmrat A

340111




340111







  • 1




    $begingroup$
    Isn't $G=mathbbR$ and $H=mathbbZ$ an example of the non-iso you want? All you need to show is that $mathbbR$ is not iso to $S^1 times mathbbZ$. That's easy.
    $endgroup$
    – Randall
    2 hours ago













  • 1




    $begingroup$
    Isn't $G=mathbbR$ and $H=mathbbZ$ an example of the non-iso you want? All you need to show is that $mathbbR$ is not iso to $S^1 times mathbbZ$. That's easy.
    $endgroup$
    – Randall
    2 hours ago








1




1




$begingroup$
Isn't $G=mathbbR$ and $H=mathbbZ$ an example of the non-iso you want? All you need to show is that $mathbbR$ is not iso to $S^1 times mathbbZ$. That's easy.
$endgroup$
– Randall
2 hours ago





$begingroup$
Isn't $G=mathbbR$ and $H=mathbbZ$ an example of the non-iso you want? All you need to show is that $mathbbR$ is not iso to $S^1 times mathbbZ$. That's easy.
$endgroup$
– Randall
2 hours ago











2 Answers
2






active

oldest

votes


















1












$begingroup$

EDIT: To be clear I was doing the case when $H$ was assumed connected. The disconnected case is handled below by Randall.



Every connected real abelian Lie group $G$ is isomorphic to $mathbbR^mtimes (S^1)^n$ for some $n$. In fact, given $G$ you can read off $n$ and $m$ as $n=mathrmrank(pi_1(G))$ and $m=dim G-n$.



Now, if you have a short exact sequence of abelian Lie groups



$$0to Hto Gto G/Hto 0$$



Then evidentily $dim G=dim H+dim G/H$. Moreover, since this is fibration, the groups are connected, and have vanishing second homotopy groups you also get a short exact sequence



$$0to pi_1(H)topi_1(G)topi_1(G/H)to 0$$



So, $mathrmrank(pi_1(G))=mathrmrank(pi_1(H))+mathrmrank(pi_1(G/H))$. Combining these two gives that $Gcong Htimes G/H$ as desired



EDIT: Here are more details. To show that $Gcong Htimes (G/H)$ it suffices to show that



$$mathrmrk(pi_1(G))=mathrmrk(pi_1(Htimes (G/H))=mathrmrk(pi_1(G))+mathrmrk(pi_1(G/H))$$



and



$$mathrmdim(G)-mathrmrk(pi_1(G))=dim(Gtimes (G/H))-mathrmrk(pi_1(Htimes (G/H))$$



The first equality holds by remark about the long exact sequence on homotopy groups from the fibration. The second is given as follows:



$$beginaligneddim(G)-mathrmrk(pi_1(G)) &= dim(H)+dim(G/H)-(mathrmrk(pi_1(H))+mathrmrk(pi_1(G/H))\ &= dim(Gtimes G/H))-mathrmrank(pi_1(Gtimes (G/H)))endaligned$$




(Below is for the non-abelian situation)
Here's a simple interesting example.



Take $mathrmGL_2(mathbbC)$ with its center $Z:=lambda I_2:lambdainmathbbC^times$. Then, $mathrmGL_2(mathbbC)/Zcong mathrmPGL_2(mathbbC)$. To see that $mathrmGL_2(mathbbC)notcong ZtimesmathrmPGL_2(mathbbC)$ note that the derived (i.e. commutative) subgroup of the former is $mathrmSL_2(mathbbC)$ whereas the latter is $mathrmPGL_2(mathbbC)$. Of course, these groups aren't isomorphic as the former is simply connected and the latter is not.






share|cite|improve this answer











$endgroup$












  • $begingroup$
    Thank you very much Alex. So the bundle $Gto G/H$ is always trivial!
    $endgroup$
    – Amrat A
    3 hours ago











  • $begingroup$
    @AmratA No problem. Did you see the updated affirmative answer to the abelian situation?
    $endgroup$
    – Alex Youcis
    3 hours ago










  • $begingroup$
    Oh yes, I just did. Thanks again!
    $endgroup$
    – Amrat A
    3 hours ago






  • 1




    $begingroup$
    @AmratA This is not true. Be careful, I didn't even necessarily claim that the fibration is trivial in my proof. I just proved that abstractly $Gcong Htimes (G/H)$, not that the sequence splits.
    $endgroup$
    – Alex Youcis
    3 hours ago






  • 1




    $begingroup$
    @AmratA Updated.
    $endgroup$
    – Alex Youcis
    3 hours ago


















3












$begingroup$

Take $G = mathbbR$ and $H=mathbbZ$. The quotient $G/H$ is the circle $S^1$. The question is now to compare $mathbbR$ to $S^1 times mathbbZ$. Now, whether you interpret $ncong$ as "not topologically iso" or "not group iso" doesn't matter, as this is a counterexample to both at once. Topologically they are distinct as $mathbbR$ is connected but $S^1 times mathbbZ$ is not (it's a stack of circles). Algebraically they're also distinct by looking at elements of order $2$ ($mathbbR$ has none, $S^1 times mathbbZ$ has at least one).






share|cite|improve this answer









$endgroup$








  • 2




    $begingroup$
    As I said in response to your comment, I generally only think about connected groups, so made that assumption (perhaps unfairly for the OP). The disconnected case as you've mentioned is quite obviously no. I edited my post to reflect this.
    $endgroup$
    – Alex Youcis
    1 hour ago











Your Answer





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2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes









1












$begingroup$

EDIT: To be clear I was doing the case when $H$ was assumed connected. The disconnected case is handled below by Randall.



Every connected real abelian Lie group $G$ is isomorphic to $mathbbR^mtimes (S^1)^n$ for some $n$. In fact, given $G$ you can read off $n$ and $m$ as $n=mathrmrank(pi_1(G))$ and $m=dim G-n$.



Now, if you have a short exact sequence of abelian Lie groups



$$0to Hto Gto G/Hto 0$$



Then evidentily $dim G=dim H+dim G/H$. Moreover, since this is fibration, the groups are connected, and have vanishing second homotopy groups you also get a short exact sequence



$$0to pi_1(H)topi_1(G)topi_1(G/H)to 0$$



So, $mathrmrank(pi_1(G))=mathrmrank(pi_1(H))+mathrmrank(pi_1(G/H))$. Combining these two gives that $Gcong Htimes G/H$ as desired



EDIT: Here are more details. To show that $Gcong Htimes (G/H)$ it suffices to show that



$$mathrmrk(pi_1(G))=mathrmrk(pi_1(Htimes (G/H))=mathrmrk(pi_1(G))+mathrmrk(pi_1(G/H))$$



and



$$mathrmdim(G)-mathrmrk(pi_1(G))=dim(Gtimes (G/H))-mathrmrk(pi_1(Htimes (G/H))$$



The first equality holds by remark about the long exact sequence on homotopy groups from the fibration. The second is given as follows:



$$beginaligneddim(G)-mathrmrk(pi_1(G)) &= dim(H)+dim(G/H)-(mathrmrk(pi_1(H))+mathrmrk(pi_1(G/H))\ &= dim(Gtimes G/H))-mathrmrank(pi_1(Gtimes (G/H)))endaligned$$




(Below is for the non-abelian situation)
Here's a simple interesting example.



Take $mathrmGL_2(mathbbC)$ with its center $Z:=lambda I_2:lambdainmathbbC^times$. Then, $mathrmGL_2(mathbbC)/Zcong mathrmPGL_2(mathbbC)$. To see that $mathrmGL_2(mathbbC)notcong ZtimesmathrmPGL_2(mathbbC)$ note that the derived (i.e. commutative) subgroup of the former is $mathrmSL_2(mathbbC)$ whereas the latter is $mathrmPGL_2(mathbbC)$. Of course, these groups aren't isomorphic as the former is simply connected and the latter is not.






share|cite|improve this answer











$endgroup$












  • $begingroup$
    Thank you very much Alex. So the bundle $Gto G/H$ is always trivial!
    $endgroup$
    – Amrat A
    3 hours ago











  • $begingroup$
    @AmratA No problem. Did you see the updated affirmative answer to the abelian situation?
    $endgroup$
    – Alex Youcis
    3 hours ago










  • $begingroup$
    Oh yes, I just did. Thanks again!
    $endgroup$
    – Amrat A
    3 hours ago






  • 1




    $begingroup$
    @AmratA This is not true. Be careful, I didn't even necessarily claim that the fibration is trivial in my proof. I just proved that abstractly $Gcong Htimes (G/H)$, not that the sequence splits.
    $endgroup$
    – Alex Youcis
    3 hours ago






  • 1




    $begingroup$
    @AmratA Updated.
    $endgroup$
    – Alex Youcis
    3 hours ago















1












$begingroup$

EDIT: To be clear I was doing the case when $H$ was assumed connected. The disconnected case is handled below by Randall.



Every connected real abelian Lie group $G$ is isomorphic to $mathbbR^mtimes (S^1)^n$ for some $n$. In fact, given $G$ you can read off $n$ and $m$ as $n=mathrmrank(pi_1(G))$ and $m=dim G-n$.



Now, if you have a short exact sequence of abelian Lie groups



$$0to Hto Gto G/Hto 0$$



Then evidentily $dim G=dim H+dim G/H$. Moreover, since this is fibration, the groups are connected, and have vanishing second homotopy groups you also get a short exact sequence



$$0to pi_1(H)topi_1(G)topi_1(G/H)to 0$$



So, $mathrmrank(pi_1(G))=mathrmrank(pi_1(H))+mathrmrank(pi_1(G/H))$. Combining these two gives that $Gcong Htimes G/H$ as desired



EDIT: Here are more details. To show that $Gcong Htimes (G/H)$ it suffices to show that



$$mathrmrk(pi_1(G))=mathrmrk(pi_1(Htimes (G/H))=mathrmrk(pi_1(G))+mathrmrk(pi_1(G/H))$$



and



$$mathrmdim(G)-mathrmrk(pi_1(G))=dim(Gtimes (G/H))-mathrmrk(pi_1(Htimes (G/H))$$



The first equality holds by remark about the long exact sequence on homotopy groups from the fibration. The second is given as follows:



$$beginaligneddim(G)-mathrmrk(pi_1(G)) &= dim(H)+dim(G/H)-(mathrmrk(pi_1(H))+mathrmrk(pi_1(G/H))\ &= dim(Gtimes G/H))-mathrmrank(pi_1(Gtimes (G/H)))endaligned$$




(Below is for the non-abelian situation)
Here's a simple interesting example.



Take $mathrmGL_2(mathbbC)$ with its center $Z:=lambda I_2:lambdainmathbbC^times$. Then, $mathrmGL_2(mathbbC)/Zcong mathrmPGL_2(mathbbC)$. To see that $mathrmGL_2(mathbbC)notcong ZtimesmathrmPGL_2(mathbbC)$ note that the derived (i.e. commutative) subgroup of the former is $mathrmSL_2(mathbbC)$ whereas the latter is $mathrmPGL_2(mathbbC)$. Of course, these groups aren't isomorphic as the former is simply connected and the latter is not.






share|cite|improve this answer











$endgroup$












  • $begingroup$
    Thank you very much Alex. So the bundle $Gto G/H$ is always trivial!
    $endgroup$
    – Amrat A
    3 hours ago











  • $begingroup$
    @AmratA No problem. Did you see the updated affirmative answer to the abelian situation?
    $endgroup$
    – Alex Youcis
    3 hours ago










  • $begingroup$
    Oh yes, I just did. Thanks again!
    $endgroup$
    – Amrat A
    3 hours ago






  • 1




    $begingroup$
    @AmratA This is not true. Be careful, I didn't even necessarily claim that the fibration is trivial in my proof. I just proved that abstractly $Gcong Htimes (G/H)$, not that the sequence splits.
    $endgroup$
    – Alex Youcis
    3 hours ago






  • 1




    $begingroup$
    @AmratA Updated.
    $endgroup$
    – Alex Youcis
    3 hours ago













1












1








1





$begingroup$

EDIT: To be clear I was doing the case when $H$ was assumed connected. The disconnected case is handled below by Randall.



Every connected real abelian Lie group $G$ is isomorphic to $mathbbR^mtimes (S^1)^n$ for some $n$. In fact, given $G$ you can read off $n$ and $m$ as $n=mathrmrank(pi_1(G))$ and $m=dim G-n$.



Now, if you have a short exact sequence of abelian Lie groups



$$0to Hto Gto G/Hto 0$$



Then evidentily $dim G=dim H+dim G/H$. Moreover, since this is fibration, the groups are connected, and have vanishing second homotopy groups you also get a short exact sequence



$$0to pi_1(H)topi_1(G)topi_1(G/H)to 0$$



So, $mathrmrank(pi_1(G))=mathrmrank(pi_1(H))+mathrmrank(pi_1(G/H))$. Combining these two gives that $Gcong Htimes G/H$ as desired



EDIT: Here are more details. To show that $Gcong Htimes (G/H)$ it suffices to show that



$$mathrmrk(pi_1(G))=mathrmrk(pi_1(Htimes (G/H))=mathrmrk(pi_1(G))+mathrmrk(pi_1(G/H))$$



and



$$mathrmdim(G)-mathrmrk(pi_1(G))=dim(Gtimes (G/H))-mathrmrk(pi_1(Htimes (G/H))$$



The first equality holds by remark about the long exact sequence on homotopy groups from the fibration. The second is given as follows:



$$beginaligneddim(G)-mathrmrk(pi_1(G)) &= dim(H)+dim(G/H)-(mathrmrk(pi_1(H))+mathrmrk(pi_1(G/H))\ &= dim(Gtimes G/H))-mathrmrank(pi_1(Gtimes (G/H)))endaligned$$




(Below is for the non-abelian situation)
Here's a simple interesting example.



Take $mathrmGL_2(mathbbC)$ with its center $Z:=lambda I_2:lambdainmathbbC^times$. Then, $mathrmGL_2(mathbbC)/Zcong mathrmPGL_2(mathbbC)$. To see that $mathrmGL_2(mathbbC)notcong ZtimesmathrmPGL_2(mathbbC)$ note that the derived (i.e. commutative) subgroup of the former is $mathrmSL_2(mathbbC)$ whereas the latter is $mathrmPGL_2(mathbbC)$. Of course, these groups aren't isomorphic as the former is simply connected and the latter is not.






share|cite|improve this answer











$endgroup$



EDIT: To be clear I was doing the case when $H$ was assumed connected. The disconnected case is handled below by Randall.



Every connected real abelian Lie group $G$ is isomorphic to $mathbbR^mtimes (S^1)^n$ for some $n$. In fact, given $G$ you can read off $n$ and $m$ as $n=mathrmrank(pi_1(G))$ and $m=dim G-n$.



Now, if you have a short exact sequence of abelian Lie groups



$$0to Hto Gto G/Hto 0$$



Then evidentily $dim G=dim H+dim G/H$. Moreover, since this is fibration, the groups are connected, and have vanishing second homotopy groups you also get a short exact sequence



$$0to pi_1(H)topi_1(G)topi_1(G/H)to 0$$



So, $mathrmrank(pi_1(G))=mathrmrank(pi_1(H))+mathrmrank(pi_1(G/H))$. Combining these two gives that $Gcong Htimes G/H$ as desired



EDIT: Here are more details. To show that $Gcong Htimes (G/H)$ it suffices to show that



$$mathrmrk(pi_1(G))=mathrmrk(pi_1(Htimes (G/H))=mathrmrk(pi_1(G))+mathrmrk(pi_1(G/H))$$



and



$$mathrmdim(G)-mathrmrk(pi_1(G))=dim(Gtimes (G/H))-mathrmrk(pi_1(Htimes (G/H))$$



The first equality holds by remark about the long exact sequence on homotopy groups from the fibration. The second is given as follows:



$$beginaligneddim(G)-mathrmrk(pi_1(G)) &= dim(H)+dim(G/H)-(mathrmrk(pi_1(H))+mathrmrk(pi_1(G/H))\ &= dim(Gtimes G/H))-mathrmrank(pi_1(Gtimes (G/H)))endaligned$$




(Below is for the non-abelian situation)
Here's a simple interesting example.



Take $mathrmGL_2(mathbbC)$ with its center $Z:=lambda I_2:lambdainmathbbC^times$. Then, $mathrmGL_2(mathbbC)/Zcong mathrmPGL_2(mathbbC)$. To see that $mathrmGL_2(mathbbC)notcong ZtimesmathrmPGL_2(mathbbC)$ note that the derived (i.e. commutative) subgroup of the former is $mathrmSL_2(mathbbC)$ whereas the latter is $mathrmPGL_2(mathbbC)$. Of course, these groups aren't isomorphic as the former is simply connected and the latter is not.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 1 hour ago

























answered 3 hours ago









Alex YoucisAlex Youcis

36k775115




36k775115











  • $begingroup$
    Thank you very much Alex. So the bundle $Gto G/H$ is always trivial!
    $endgroup$
    – Amrat A
    3 hours ago











  • $begingroup$
    @AmratA No problem. Did you see the updated affirmative answer to the abelian situation?
    $endgroup$
    – Alex Youcis
    3 hours ago










  • $begingroup$
    Oh yes, I just did. Thanks again!
    $endgroup$
    – Amrat A
    3 hours ago






  • 1




    $begingroup$
    @AmratA This is not true. Be careful, I didn't even necessarily claim that the fibration is trivial in my proof. I just proved that abstractly $Gcong Htimes (G/H)$, not that the sequence splits.
    $endgroup$
    – Alex Youcis
    3 hours ago






  • 1




    $begingroup$
    @AmratA Updated.
    $endgroup$
    – Alex Youcis
    3 hours ago
















  • $begingroup$
    Thank you very much Alex. So the bundle $Gto G/H$ is always trivial!
    $endgroup$
    – Amrat A
    3 hours ago











  • $begingroup$
    @AmratA No problem. Did you see the updated affirmative answer to the abelian situation?
    $endgroup$
    – Alex Youcis
    3 hours ago










  • $begingroup$
    Oh yes, I just did. Thanks again!
    $endgroup$
    – Amrat A
    3 hours ago






  • 1




    $begingroup$
    @AmratA This is not true. Be careful, I didn't even necessarily claim that the fibration is trivial in my proof. I just proved that abstractly $Gcong Htimes (G/H)$, not that the sequence splits.
    $endgroup$
    – Alex Youcis
    3 hours ago






  • 1




    $begingroup$
    @AmratA Updated.
    $endgroup$
    – Alex Youcis
    3 hours ago















$begingroup$
Thank you very much Alex. So the bundle $Gto G/H$ is always trivial!
$endgroup$
– Amrat A
3 hours ago





$begingroup$
Thank you very much Alex. So the bundle $Gto G/H$ is always trivial!
$endgroup$
– Amrat A
3 hours ago













$begingroup$
@AmratA No problem. Did you see the updated affirmative answer to the abelian situation?
$endgroup$
– Alex Youcis
3 hours ago




$begingroup$
@AmratA No problem. Did you see the updated affirmative answer to the abelian situation?
$endgroup$
– Alex Youcis
3 hours ago












$begingroup$
Oh yes, I just did. Thanks again!
$endgroup$
– Amrat A
3 hours ago




$begingroup$
Oh yes, I just did. Thanks again!
$endgroup$
– Amrat A
3 hours ago




1




1




$begingroup$
@AmratA This is not true. Be careful, I didn't even necessarily claim that the fibration is trivial in my proof. I just proved that abstractly $Gcong Htimes (G/H)$, not that the sequence splits.
$endgroup$
– Alex Youcis
3 hours ago




$begingroup$
@AmratA This is not true. Be careful, I didn't even necessarily claim that the fibration is trivial in my proof. I just proved that abstractly $Gcong Htimes (G/H)$, not that the sequence splits.
$endgroup$
– Alex Youcis
3 hours ago




1




1




$begingroup$
@AmratA Updated.
$endgroup$
– Alex Youcis
3 hours ago




$begingroup$
@AmratA Updated.
$endgroup$
– Alex Youcis
3 hours ago











3












$begingroup$

Take $G = mathbbR$ and $H=mathbbZ$. The quotient $G/H$ is the circle $S^1$. The question is now to compare $mathbbR$ to $S^1 times mathbbZ$. Now, whether you interpret $ncong$ as "not topologically iso" or "not group iso" doesn't matter, as this is a counterexample to both at once. Topologically they are distinct as $mathbbR$ is connected but $S^1 times mathbbZ$ is not (it's a stack of circles). Algebraically they're also distinct by looking at elements of order $2$ ($mathbbR$ has none, $S^1 times mathbbZ$ has at least one).






share|cite|improve this answer









$endgroup$








  • 2




    $begingroup$
    As I said in response to your comment, I generally only think about connected groups, so made that assumption (perhaps unfairly for the OP). The disconnected case as you've mentioned is quite obviously no. I edited my post to reflect this.
    $endgroup$
    – Alex Youcis
    1 hour ago















3












$begingroup$

Take $G = mathbbR$ and $H=mathbbZ$. The quotient $G/H$ is the circle $S^1$. The question is now to compare $mathbbR$ to $S^1 times mathbbZ$. Now, whether you interpret $ncong$ as "not topologically iso" or "not group iso" doesn't matter, as this is a counterexample to both at once. Topologically they are distinct as $mathbbR$ is connected but $S^1 times mathbbZ$ is not (it's a stack of circles). Algebraically they're also distinct by looking at elements of order $2$ ($mathbbR$ has none, $S^1 times mathbbZ$ has at least one).






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$endgroup$








  • 2




    $begingroup$
    As I said in response to your comment, I generally only think about connected groups, so made that assumption (perhaps unfairly for the OP). The disconnected case as you've mentioned is quite obviously no. I edited my post to reflect this.
    $endgroup$
    – Alex Youcis
    1 hour ago













3












3








3





$begingroup$

Take $G = mathbbR$ and $H=mathbbZ$. The quotient $G/H$ is the circle $S^1$. The question is now to compare $mathbbR$ to $S^1 times mathbbZ$. Now, whether you interpret $ncong$ as "not topologically iso" or "not group iso" doesn't matter, as this is a counterexample to both at once. Topologically they are distinct as $mathbbR$ is connected but $S^1 times mathbbZ$ is not (it's a stack of circles). Algebraically they're also distinct by looking at elements of order $2$ ($mathbbR$ has none, $S^1 times mathbbZ$ has at least one).






share|cite|improve this answer









$endgroup$



Take $G = mathbbR$ and $H=mathbbZ$. The quotient $G/H$ is the circle $S^1$. The question is now to compare $mathbbR$ to $S^1 times mathbbZ$. Now, whether you interpret $ncong$ as "not topologically iso" or "not group iso" doesn't matter, as this is a counterexample to both at once. Topologically they are distinct as $mathbbR$ is connected but $S^1 times mathbbZ$ is not (it's a stack of circles). Algebraically they're also distinct by looking at elements of order $2$ ($mathbbR$ has none, $S^1 times mathbbZ$ has at least one).







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 1 hour ago









RandallRandall

10.7k11431




10.7k11431







  • 2




    $begingroup$
    As I said in response to your comment, I generally only think about connected groups, so made that assumption (perhaps unfairly for the OP). The disconnected case as you've mentioned is quite obviously no. I edited my post to reflect this.
    $endgroup$
    – Alex Youcis
    1 hour ago












  • 2




    $begingroup$
    As I said in response to your comment, I generally only think about connected groups, so made that assumption (perhaps unfairly for the OP). The disconnected case as you've mentioned is quite obviously no. I edited my post to reflect this.
    $endgroup$
    – Alex Youcis
    1 hour ago







2




2




$begingroup$
As I said in response to your comment, I generally only think about connected groups, so made that assumption (perhaps unfairly for the OP). The disconnected case as you've mentioned is quite obviously no. I edited my post to reflect this.
$endgroup$
– Alex Youcis
1 hour ago




$begingroup$
As I said in response to your comment, I generally only think about connected groups, so made that assumption (perhaps unfairly for the OP). The disconnected case as you've mentioned is quite obviously no. I edited my post to reflect this.
$endgroup$
– Alex Youcis
1 hour ago

















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