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It is well known that every subgroup $H$ of a finite abelian group $G$ is isomorphic to a quotient of $G$. I'm wondering whether there is a counterpart for profinite groups.

For example is it true that every open subgroup $H$ of an abelian profinite group $G$ isomorphic to a quotient?

what about the weaker statement that every open subgroup contains an open subgroup which is isomorphic to a quotient?

If G is a direct product of finite groups then the second weaker statement is true because every open subgroup $H$ contains a cylinder neighborhood of the origin $H'$ for which $G=H'\times K$ where $K$ is a finite product of finite groups.

Edit: (This is wrong) Ok I believe I proved the second weaker statement. Let $G$ be a profinite group, then $G$ embeds into a direct product of finite groups $\Gamma$. Let $H$ be an open subgroup of $G$, therefore it is of finite index. So there exists an open subgroup $U\leq \Gamma$ so that $U\cap G\subseteq H$ (Here's a link!). Now $U$ contains an open subgroup $W$ so that $\Gamma =W\times K$ Consider the set $W\cap G$ we have a map $\phi:G\rightarrow \Gamma\rightarrow \Gamma/K=W$ This map is onto $W\cap G$ (it is onto the set of all $w\in W$ so that there exists $k\in K$ so that $(w,k)\in G$) It follows that $W\cap G$ is a subgroup of $H$ that is isomorphic to a quotient.

It is well known that every subgroup $H$ of a finite abelian group $G$ is isomorphic to a quotient of $G$. I'm wondering whether there is a counterpart for profinite groups.

For example is it true that every open subgroup $H$ of an abelian profinite group $G$ isomorphic to a quotient?

what about the weaker statement that every open subgroup contains an open subgroup which is isomorphic to a quotient?

If G is a direct product of finite groups then the second weaker statement is true because every open subgroup $H$ contains a cylinder neighborhood of the origin $H'$ for which $G=H'\times K$ where $K$ is a finite product of finite groups.

Thanks

It is well known that every subgroup $H$ of a finite abelian group $G$ is isomorphic to a quotient of $G$. I'm wondering whether there is a counterpart for profinite groups.

For example is it true that every open subgroup $H$ of an abelian profinite group $G$ isomorphic to a quotient?

what about the weaker statement that every open subgroup contains an open subgroup which is isomorphic to a quotient?

If G is a direct product of finite groups then the second weaker statement is true because every open subgroup $H$ contains a cylinder neighborhood of the origin $H'$ for which $G=H'\times K$ where $K$ is a finite product of finite groups.

Edit: (This is wrong) Ok I believe i can prove the second weaker statement. Let $G$ be a profinite group, then $G$ embeds into a direct product of finite groups $\Gamma$. Let $H$ be an open subgroup of $G$, therefore it is of finite index. So there exists an open subgroup $U\leq \Gamma$ so that $U\cap G\subseteq H$ (Here's a link!). Now $U$ contains an open subgroup $W$ so that $\Gamma =W\times K$ Consider the set $W\cap G$ we have a map $\phi:G\rightarrow \Gamma\rightarrow \Gamma/K=W$ This map is onto $W\cap G$ (it is onto the set of all $w\in W$ so that there exists $k\in K$ so that $(w,k)\in G$) It follows that $W\cap G$ is a subgroup of $H$ that is isomorphic to a quotient.

Please write a comment if you find a mistake.

It is well known that every subgroup $H$ of a finite abelian group $G$ is isomorphic to a quotient of $G$. I'm wondering whether there is a counterpart for profinite groups.

For example is it true that every open subgroup $H$ of an abelian profinite group $G$ isomorphic to a quotient?

what about the weaker statement that every open subgroup contains an open subgroup which is isomorphic to a quotient?

If G is a direct product of finite groups then the second weaker statement is true because every open subgroup $H$ contains a cylinder neighborhood of the origin $H'$ for which $G=H'\times K$ where $K$ is a finite product of finite groups.

Edit: (This is wrong) Ok I believe I proved the second weaker statement. Let $G$ be a profinite group, then $G$ embeds into a direct product of finite groups $\Gamma$. Let $H$ be an open subgroup of $G$, therefore it is of finite index. So there exists an open subgroup $U\leq \Gamma$ so that $U\cap G\subseteq H$ (Here's a link!). Now $U$ contains an open subgroup $W$ so that $\Gamma =W\times K$ Consider the set $W\cap G$ we have a map $\phi:G\rightarrow \Gamma\rightarrow \Gamma/K=W$ This map is onto $W\cap G$ (it is onto the set of all $w\in W$ so that there exists $k\in K$ so that $(w,k)\in G$) It follows that $W\cap G$ is a subgroup of $H$ that is isomorphic to a quotient.

It is well known that every subgroup $H$ of a finite abelian group $G$ is isomorphic to a quotient of $G$. I'm wondering whether there is a counterpart for profinite groups.

For example is it true that every open subgroup $H$ of an abelian profinite group $G$ isomorphic to a quotient?

what about the weaker statement that every open subgroup contains an open subgroup which is isomorphic to a quotient?

If G is a direct product of finite groups then the second weaker statement is true because every open subgroup $H$ contains a cylinder neighborhood of the origin $H'$ for which $G=H'\times K$ where $K$ is a finite product of finite groups.

Edit: Ok I believe i can prove the second weaker statement. Let $G$ be a profinite group, then $G$ embeds into a direct product of finite groups $\Gamma$. Let $H$ be an open subgroup of $G$, therefore it is of finite index. So there exists an open subgroup $U\leq \Gamma$ so that $U\cap G\subseteq H$ (Here's a link!). Now $U$ contains an open subgroup $W$ so that $\Gamma =W\times K$ Consider the set $W\cap G$ we have a map $\phi:G\rightarrow \Gamma\rightarrow \Gamma/K=W$ This map is onto $W\cap G$ It follows that $W\cap G$ is a subgroup of $H$ that is isomorphic to a quotient.

Please write a comment if you find a mistake.

It is well known that every subgroup $H$ of a finite abelian group $G$ is isomorphic to a quotient of $G$. I'm wondering whether there is a counterpart for profinite groups.

For example is it true that every open subgroup $H$ of an abelian profinite group $G$ isomorphic to a quotient?

what about the weaker statement that every open subgroup contains an open subgroup which is isomorphic to a quotient?

If G is a direct product of finite groups then the second weaker statement is true because every open subgroup $H$ contains a cylinder neighborhood of the origin $H'$ for which $G=H'\times K$ where $K$ is a finite product of finite groups.

Edit: (This is wrong) Ok I believe i can prove the second weaker statement. Let $G$ be a profinite group, then $G$ embeds into a direct product of finite groups $\Gamma$. Let $H$ be an open subgroup of $G$, therefore it is of finite index. So there exists an open subgroup $U\leq \Gamma$ so that $U\cap G\subseteq H$ (Here's a link!). Now $U$ contains an open subgroup $W$ so that $\Gamma =W\times K$ Consider the set $W\cap G$ we have a map $\phi:G\rightarrow \Gamma\rightarrow \Gamma/K=W$ This map is onto $W\cap G$ (it is onto the set of all $w\in W$ so that there exists $k\in K$ so that $(w,k)\in G$) It follows that $W\cap G$ is a subgroup of $H$ that is isomorphic to a quotient.

Please write a comment if you find a mistake.