Let $G$ be a finite group with a normal subgroup $H$ such that $G/H$ has prime order . Then precisely when $G$ is isomorphic to $ H × G/H$ .

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Let $G$ be a finite group with a normal subgroup $H$ . Then precisely when $G$ is isomorphic to $ H × G/H$ .



I was trying to directly construct an isomorphism with trial and error between $G$ and $ H × G/H$ and my hope was to find appropriate conditions on $G$ and $H$ along the way.But so far not able to make any notable progress.



I was trying different onto homomorphisms from $Gto H$ and $Gto G/H$ , and then use these homomorphisms as the first and the second component of the desired isomorphism between $G$ and $ H × G/H$. I was trying to see how those homomorphisms should relate to each other in order to do the job and what constrain they put on $G$ and $H$ ?



Any progress in this situation will be help full to me .Thanks




abstract-algebra group-theory finite-groups




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  • It is not very clear to me, what are you asking. Please give more details.
    – peterh
    Jul 15 at 15:44






  • 1




    This is a very broad question. One characterization is when there is a projection $pi:Gto H$ which is the identity on $H$.
    – Ashwin Trisal
    Jul 15 at 15:47











  • A necessary (but not sufficient) condition is that $H$ is in the center of $G$.
    – Andreas Blass
    Jul 15 at 17:45














up vote
2
down vote

favorite












Let $G$ be a finite group with a normal subgroup $H$ . Then precisely when $G$ is isomorphic to $ H × G/H$ .



I was trying to directly construct an isomorphism with trial and error between $G$ and $ H × G/H$ and my hope was to find appropriate conditions on $G$ and $H$ along the way.But so far not able to make any notable progress.



I was trying different onto homomorphisms from $Gto H$ and $Gto G/H$ , and then use these homomorphisms as the first and the second component of the desired isomorphism between $G$ and $ H × G/H$. I was trying to see how those homomorphisms should relate to each other in order to do the job and what constrain they put on $G$ and $H$ ?



Any progress in this situation will be help full to me .Thanks




abstract-algebra group-theory finite-groups




share|cite|improve this question





















  • It is not very clear to me, what are you asking. Please give more details.
    – peterh
    Jul 15 at 15:44






  • 1




    This is a very broad question. One characterization is when there is a projection $pi:Gto H$ which is the identity on $H$.
    – Ashwin Trisal
    Jul 15 at 15:47











  • A necessary (but not sufficient) condition is that $H$ is in the center of $G$.
    – Andreas Blass
    Jul 15 at 17:45












up vote
2
down vote

favorite









up vote
2
down vote

favorite











Let $G$ be a finite group with a normal subgroup $H$ . Then precisely when $G$ is isomorphic to $ H × G/H$ .



I was trying to directly construct an isomorphism with trial and error between $G$ and $ H × G/H$ and my hope was to find appropriate conditions on $G$ and $H$ along the way.But so far not able to make any notable progress.



I was trying different onto homomorphisms from $Gto H$ and $Gto G/H$ , and then use these homomorphisms as the first and the second component of the desired isomorphism between $G$ and $ H × G/H$. I was trying to see how those homomorphisms should relate to each other in order to do the job and what constrain they put on $G$ and $H$ ?



Any progress in this situation will be help full to me .Thanks




abstract-algebra group-theory finite-groups




share|cite|improve this question













Let $G$ be a finite group with a normal subgroup $H$ . Then precisely when $G$ is isomorphic to $ H × G/H$ .



I was trying to directly construct an isomorphism with trial and error between $G$ and $ H × G/H$ and my hope was to find appropriate conditions on $G$ and $H$ along the way.But so far not able to make any notable progress.



I was trying different onto homomorphisms from $Gto H$ and $Gto G/H$ , and then use these homomorphisms as the first and the second component of the desired isomorphism between $G$ and $ H × G/H$. I was trying to see how those homomorphisms should relate to each other in order to do the job and what constrain they put on $G$ and $H$ ?



Any progress in this situation will be help full to me .Thanks





abstract-algebra group-theory finite-groups





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 16 at 11:38
























asked Jul 15 at 15:07









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  • It is not very clear to me, what are you asking. Please give more details.
    – peterh
    Jul 15 at 15:44






  • 1




    This is a very broad question. One characterization is when there is a projection $pi:Gto H$ which is the identity on $H$.
    – Ashwin Trisal
    Jul 15 at 15:47











  • A necessary (but not sufficient) condition is that $H$ is in the center of $G$.
    – Andreas Blass
    Jul 15 at 17:45
















  • It is not very clear to me, what are you asking. Please give more details.
    – peterh
    Jul 15 at 15:44






  • 1




    This is a very broad question. One characterization is when there is a projection $pi:Gto H$ which is the identity on $H$.
    – Ashwin Trisal
    Jul 15 at 15:47











  • A necessary (but not sufficient) condition is that $H$ is in the center of $G$.
    – Andreas Blass
    Jul 15 at 17:45















It is not very clear to me, what are you asking. Please give more details.
– peterh
Jul 15 at 15:44




It is not very clear to me, what are you asking. Please give more details.
– peterh
Jul 15 at 15:44




1




1




This is a very broad question. One characterization is when there is a projection $pi:Gto H$ which is the identity on $H$.
– Ashwin Trisal
Jul 15 at 15:47





This is a very broad question. One characterization is when there is a projection $pi:Gto H$ which is the identity on $H$.
– Ashwin Trisal
Jul 15 at 15:47













A necessary (but not sufficient) condition is that $H$ is in the center of $G$.
– Andreas Blass
Jul 15 at 17:45




A necessary (but not sufficient) condition is that $H$ is in the center of $G$.
– Andreas Blass
Jul 15 at 17:45















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