What is the profinite completion of a free abelian group of infinite rank?
$begingroup$
By definition, profinite completion of a group $G$ is $widehat{G}=varprojlim_N G/N$ where $N$ runs through every subgroup of finite index in $G$.
Let $M=bigoplus_{nge1} Bbb{Z}$ be a free abelian group of countably infinite rank.
$1$. What is $widehat{M}$?
My guess is $widehat{M}=prod_{nge1}Bbb{widehat{Z}}$.
Am I right? How can I prove?
$2$. More generally, what is $widehat{oplus_{nge1}{ C_n}}$ where $C_n$ is cyclic group? Is it ${prod_{nge1}{widehat{ C_n}}}$?
Similarly what is pro-$p$-completions?
My questioins are originated from the profinite completion of $Bbb{Q}^{times}$, the multiplicative group of the rational number field.
It is known that $Bbb{Q}^{times}cong {{pm1}}times bigoplus_{nge1} Bbb{Z} $
group-theory topological-groups profinite-groups
edited Jan 15 at 2:31
Andrews
3901317
asked Jan 14 at 5:51
MiRi_NaEMiRi_NaE
8910
$endgroup$
add a comment |
$begingroup$
By definition, profinite completion of a group $G$ is $widehat{G}=varprojlim_N G/N$ where $N$ runs through every subgroup of finite index in $G$.
Let $M=bigoplus_{nge1} Bbb{Z}$ be a free abelian group of countably infinite rank.
$1$. What is $widehat{M}$?
My guess is $widehat{M}=prod_{nge1}Bbb{widehat{Z}}$.
Am I right? How can I prove?
$2$. More generally, what is $widehat{oplus_{nge1}{ C_n}}$ where $C_n$ is cyclic group? Is it ${prod_{nge1}{widehat{ C_n}}}$?
Similarly what is pro-$p$-completions?
My questioins are originated from the profinite completion of $Bbb{Q}^{times}$, the multiplicative group of the rational number field.
It is known that $Bbb{Q}^{times}cong {{pm1}}times bigoplus_{nge1} Bbb{Z} $
group-theory topological-groups profinite-groups
edited Jan 15 at 2:31
Andrews
3901317
asked Jan 14 at 5:51
MiRi_NaEMiRi_NaE
8910
$endgroup$
add a comment |
$begingroup$
By definition, profinite completion of a group $G$ is $widehat{G}=varprojlim_N G/N$ where $N$ runs through every subgroup of finite index in $G$.
Let $M=bigoplus_{nge1} Bbb{Z}$ be a free abelian group of countably infinite rank.
$1$. What is $widehat{M}$?
My guess is $widehat{M}=prod_{nge1}Bbb{widehat{Z}}$.
Am I right? How can I prove?
$2$. More generally, what is $widehat{oplus_{nge1}{ C_n}}$ where $C_n$ is cyclic group? Is it ${prod_{nge1}{widehat{ C_n}}}$?
Similarly what is pro-$p$-completions?
My questioins are originated from the profinite completion of $Bbb{Q}^{times}$, the multiplicative group of the rational number field.
It is known that $Bbb{Q}^{times}cong {{pm1}}times bigoplus_{nge1} Bbb{Z} $
group-theory topological-groups profinite-groups
edited Jan 15 at 2:31
Andrews
3901317
asked Jan 14 at 5:51
MiRi_NaEMiRi_NaE
8910
$endgroup$
By definition, profinite completion of a group $G$ is $widehat{G}=varprojlim_N G/N$ where $N$ runs through every subgroup of finite index in $G$.
Let $M=bigoplus_{nge1} Bbb{Z}$ be a free abelian group of countably infinite rank.
$1$. What is $widehat{M}$?
My guess is $widehat{M}=prod_{nge1}Bbb{widehat{Z}}$.
Am I right? How can I prove?
$2$. More generally, what is $widehat{oplus_{nge1}{ C_n}}$ where $C_n$ is cyclic group? Is it ${prod_{nge1}{widehat{ C_n}}}$?
Similarly what is pro-$p$-completions?
My questioins are originated from the profinite completion of $Bbb{Q}^{times}$, the multiplicative group of the rational number field.
It is known that $Bbb{Q}^{times}cong {{pm1}}times bigoplus_{nge1} Bbb{Z} $
group-theory topological-groups profinite-groups
group-theory topological-groups profinite-groups
edited Jan 15 at 2:31
Andrews
3901317
asked Jan 14 at 5:51
MiRi_NaEMiRi_NaE
8910
edited Jan 15 at 2:31
Andrews
3901317
asked Jan 14 at 5:51
MiRi_NaEMiRi_NaE
8910
edited Jan 15 at 2:31
Andrews
3901317
edited Jan 15 at 2:31
Andrews
3901317
edited Jan 15 at 2:31
Andrews
3901317
3901317
asked Jan 14 at 5:51
MiRi_NaEMiRi_NaE
8910
asked Jan 14 at 5:51
MiRi_NaEMiRi_NaE
8910
asked Jan 14 at 5:51
MiRi_NaEMiRi_NaE
8910
8910
add a comment |
add a comment |
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