Mixing
Split extensions of $G/U$ by elementary abelian $p$-group and $H^1(U, mathbbZ/pmathbbZ)$
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This question is a followup to Serre's Exercise 3.4.1 on Galois Cohomology. Let $G$ be a profinite group, $1 to P to E to W to 1$ a finite group extension and $fcolon G to W$ a continuous surjection. I want to show the equivalence of the two following properties:
- Whenever $P$ is an abelian group killed by $p$ and $E simeq P rtimes W$, there is a surjective lifting $f'colon G to E$ of $f$.
- For every open normal subgroup $U$ of $G$, and for any integer $N geq 0$, there exist $z_1,cdots,z_N in H^1(U, mathbbZ/pmathbbZ)$ such that the elements $s(z_i)$ ($s in G/U$, $1 leq i leq N$) are linearly independent over $mathbbZ/pmathbbZ$.
THE ATTEMPTS SO FAR I would guess this has to do with the five-term exact sequence. Also, I'm not sure if the pullback $1 to P to E_f to G to 1$ of the split extension $1 to P to E to W to 1$ also splits (though I would guess so). If this is the case, maybe I can get something out of $H^2(G, P)$.
group-cohomology semidirect-product profinite-groups group-extensions
asked Jul 31 at 18:56
Henrique Augusto Souza
1,226314
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up vote
2
down vote
favorite
This question is a followup to Serre's Exercise 3.4.1 on Galois Cohomology. Let $G$ be a profinite group, $1 to P to E to W to 1$ a finite group extension and $fcolon G to W$ a continuous surjection. I want to show the equivalence of the two following properties:
- Whenever $P$ is an abelian group killed by $p$ and $E simeq P rtimes W$, there is a surjective lifting $f'colon G to E$ of $f$.
- For every open normal subgroup $U$ of $G$, and for any integer $N geq 0$, there exist $z_1,cdots,z_N in H^1(U, mathbbZ/pmathbbZ)$ such that the elements $s(z_i)$ ($s in G/U$, $1 leq i leq N$) are linearly independent over $mathbbZ/pmathbbZ$.
THE ATTEMPTS SO FAR I would guess this has to do with the five-term exact sequence. Also, I'm not sure if the pullback $1 to P to E_f to G to 1$ of the split extension $1 to P to E to W to 1$ also splits (though I would guess so). If this is the case, maybe I can get something out of $H^2(G, P)$.
group-cohomology semidirect-product profinite-groups group-extensions
asked Jul 31 at 18:56
Henrique Augusto Souza
1,226314
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
This question is a followup to Serre's Exercise 3.4.1 on Galois Cohomology. Let $G$ be a profinite group, $1 to P to E to W to 1$ a finite group extension and $fcolon G to W$ a continuous surjection. I want to show the equivalence of the two following properties:
- Whenever $P$ is an abelian group killed by $p$ and $E simeq P rtimes W$, there is a surjective lifting $f'colon G to E$ of $f$.
- For every open normal subgroup $U$ of $G$, and for any integer $N geq 0$, there exist $z_1,cdots,z_N in H^1(U, mathbbZ/pmathbbZ)$ such that the elements $s(z_i)$ ($s in G/U$, $1 leq i leq N$) are linearly independent over $mathbbZ/pmathbbZ$.
THE ATTEMPTS SO FAR I would guess this has to do with the five-term exact sequence. Also, I'm not sure if the pullback $1 to P to E_f to G to 1$ of the split extension $1 to P to E to W to 1$ also splits (though I would guess so). If this is the case, maybe I can get something out of $H^2(G, P)$.
group-cohomology semidirect-product profinite-groups group-extensions
asked Jul 31 at 18:56
Henrique Augusto Souza
1,226314
This question is a followup to Serre's Exercise 3.4.1 on Galois Cohomology. Let $G$ be a profinite group, $1 to P to E to W to 1$ a finite group extension and $fcolon G to W$ a continuous surjection. I want to show the equivalence of the two following properties:
- Whenever $P$ is an abelian group killed by $p$ and $E simeq P rtimes W$, there is a surjective lifting $f'colon G to E$ of $f$.
- For every open normal subgroup $U$ of $G$, and for any integer $N geq 0$, there exist $z_1,cdots,z_N in H^1(U, mathbbZ/pmathbbZ)$ such that the elements $s(z_i)$ ($s in G/U$, $1 leq i leq N$) are linearly independent over $mathbbZ/pmathbbZ$.
THE ATTEMPTS SO FAR I would guess this has to do with the five-term exact sequence. Also, I'm not sure if the pullback $1 to P to E_f to G to 1$ of the split extension $1 to P to E to W to 1$ also splits (though I would guess so). If this is the case, maybe I can get something out of $H^2(G, P)$.
group-cohomology semidirect-product profinite-groups group-extensions
asked Jul 31 at 18:56
Henrique Augusto Souza
1,226314
asked Jul 31 at 18:56
Henrique Augusto Souza
1,226314
asked Jul 31 at 18:56
Henrique Augusto Souza
1,226314
asked Jul 31 at 18:56
Henrique Augusto Souza
1,226314
1,226314
add a comment |Â
add a comment |Â
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