Locally compact nilpotent group has an open subgroup isomorphic to $mathbb{R}^ntimes K$
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My question is about a possible generalization of the following structure theorem of locally compact abelian groups.
Theorem: Let $G$ be a locally compact abelian group. Then here exists a compact subgroup $K$ and a non-negative number $ninmathbb{N}$ such that $mathbb{R}^ntimes K$ is isomorphic to an open subgroup of $G$.
I wonder whether it is possible to generalize the above for non-abelian but nilpotent groups.
The most simple case is when $G$ is nilpotent of order $3$. In this case $[G,G]$ is an abelian group, hence $[G,G]$ satisfies the Theorem. Moreover $G/[G,G]$ is always an abelian group and so satisfies the Theorem.
Can we deduce that $G$ also satisfies the theorem?
topological-groups locally-compact-groups nilpotent-groups
asked Oct 20 at 12:14
Yanko
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My question is about a possible generalization of the following structure theorem of locally compact abelian groups.
Theorem: Let $G$ be a locally compact abelian group. Then here exists a compact subgroup $K$ and a non-negative number $ninmathbb{N}$ such that $mathbb{R}^ntimes K$ is isomorphic to an open subgroup of $G$.
I wonder whether it is possible to generalize the above for non-abelian but nilpotent groups.
The most simple case is when $G$ is nilpotent of order $3$. In this case $[G,G]$ is an abelian group, hence $[G,G]$ satisfies the Theorem. Moreover $G/[G,G]$ is always an abelian group and so satisfies the Theorem.
Can we deduce that $G$ also satisfies the theorem?
topological-groups locally-compact-groups nilpotent-groups
asked Oct 20 at 12:14
Yanko
5,123722
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
My question is about a possible generalization of the following structure theorem of locally compact abelian groups.
Theorem: Let $G$ be a locally compact abelian group. Then here exists a compact subgroup $K$ and a non-negative number $ninmathbb{N}$ such that $mathbb{R}^ntimes K$ is isomorphic to an open subgroup of $G$.
I wonder whether it is possible to generalize the above for non-abelian but nilpotent groups.
The most simple case is when $G$ is nilpotent of order $3$. In this case $[G,G]$ is an abelian group, hence $[G,G]$ satisfies the Theorem. Moreover $G/[G,G]$ is always an abelian group and so satisfies the Theorem.
Can we deduce that $G$ also satisfies the theorem?
topological-groups locally-compact-groups nilpotent-groups
asked Oct 20 at 12:14
Yanko
5,123722
My question is about a possible generalization of the following structure theorem of locally compact abelian groups.
Theorem: Let $G$ be a locally compact abelian group. Then here exists a compact subgroup $K$ and a non-negative number $ninmathbb{N}$ such that $mathbb{R}^ntimes K$ is isomorphic to an open subgroup of $G$.
I wonder whether it is possible to generalize the above for non-abelian but nilpotent groups.
The most simple case is when $G$ is nilpotent of order $3$. In this case $[G,G]$ is an abelian group, hence $[G,G]$ satisfies the Theorem. Moreover $G/[G,G]$ is always an abelian group and so satisfies the Theorem.
Can we deduce that $G$ also satisfies the theorem?
topological-groups locally-compact-groups nilpotent-groups
topological-groups locally-compact-groups nilpotent-groups
asked Oct 20 at 12:14
Yanko
5,123722
asked Oct 20 at 12:14
Yanko
5,123722
asked Oct 20 at 12:14
Yanko
5,123722
asked Oct 20 at 12:14
Yanko
5,123722
asked Oct 20 at 12:14
Yanko
5,123722
5,123722
add a comment |
add a comment |
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