Coset space and orbit space of group action
$begingroup$
Assume $G $ is a topological group and $H $ is its subgroup. Is it true that the right coset space, $G/H $, and the orbit space of the action of $H $ on $G $ are homeomorphic? (I consider the quotient topologies on both sets)
algebraic-topology finite-groups group-actions
asked Dec 16 '18 at 6:32
piotrmizerkapiotrmizerka
339110
$endgroup$
add a comment |
$begingroup$
Assume $G $ is a topological group and $H $ is its subgroup. Is it true that the right coset space, $G/H $, and the orbit space of the action of $H $ on $G $ are homeomorphic? (I consider the quotient topologies on both sets)
algebraic-topology finite-groups group-actions
asked Dec 16 '18 at 6:32
piotrmizerkapiotrmizerka
339110
$endgroup$
add a comment |
$begingroup$
Assume $G $ is a topological group and $H $ is its subgroup. Is it true that the right coset space, $G/H $, and the orbit space of the action of $H $ on $G $ are homeomorphic? (I consider the quotient topologies on both sets)
algebraic-topology finite-groups group-actions
asked Dec 16 '18 at 6:32
piotrmizerkapiotrmizerka
339110
$endgroup$
Assume $G $ is a topological group and $H $ is its subgroup. Is it true that the right coset space, $G/H $, and the orbit space of the action of $H $ on $G $ are homeomorphic? (I consider the quotient topologies on both sets)
algebraic-topology finite-groups group-actions
algebraic-topology finite-groups group-actions
asked Dec 16 '18 at 6:32
piotrmizerkapiotrmizerka
339110
asked Dec 16 '18 at 6:32
piotrmizerkapiotrmizerka
339110
asked Dec 16 '18 at 6:32
piotrmizerkapiotrmizerka
339110
asked Dec 16 '18 at 6:32
piotrmizerkapiotrmizerka
339110
asked Dec 16 '18 at 6:32
piotrmizerkapiotrmizerka
339110
339110
add a comment |
add a comment |
1 Answer
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I assume we are talking about the action given by $(h,g)mapsto hg$. Let $orb(G)$ denote the orbit space of $G$ under the action of $H$. So what is the orbit of $gin G$? It is simply $Hg$. Therefore literally $orb(G)=G/H$ where $G/H$ denotes the set of all right cosets. Since they are generated by the same partitioning of $G$ the quotient topologies have to agree as well.
answered Dec 16 '18 at 12:24
freakishfreakish
12.3k1630
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1 Answer
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1 Answer
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active
oldest
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$begingroup$
I assume we are talking about the action given by $(h,g)mapsto hg$. Let $orb(G)$ denote the orbit space of $G$ under the action of $H$. So what is the orbit of $gin G$? It is simply $Hg$. Therefore literally $orb(G)=G/H$ where $G/H$ denotes the set of all right cosets. Since they are generated by the same partitioning of $G$ the quotient topologies have to agree as well.
answered Dec 16 '18 at 12:24
freakishfreakish
12.3k1630
$endgroup$
add a comment |
$begingroup$
I assume we are talking about the action given by $(h,g)mapsto hg$. Let $orb(G)$ denote the orbit space of $G$ under the action of $H$. So what is the orbit of $gin G$? It is simply $Hg$. Therefore literally $orb(G)=G/H$ where $G/H$ denotes the set of all right cosets. Since they are generated by the same partitioning of $G$ the quotient topologies have to agree as well.
answered Dec 16 '18 at 12:24
freakishfreakish
12.3k1630
$endgroup$
add a comment |
$begingroup$
I assume we are talking about the action given by $(h,g)mapsto hg$. Let $orb(G)$ denote the orbit space of $G$ under the action of $H$. So what is the orbit of $gin G$? It is simply $Hg$. Therefore literally $orb(G)=G/H$ where $G/H$ denotes the set of all right cosets. Since they are generated by the same partitioning of $G$ the quotient topologies have to agree as well.
answered Dec 16 '18 at 12:24
freakishfreakish
12.3k1630
$endgroup$
I assume we are talking about the action given by $(h,g)mapsto hg$. Let $orb(G)$ denote the orbit space of $G$ under the action of $H$. So what is the orbit of $gin G$? It is simply $Hg$. Therefore literally $orb(G)=G/H$ where $G/H$ denotes the set of all right cosets. Since they are generated by the same partitioning of $G$ the quotient topologies have to agree as well.
answered Dec 16 '18 at 12:24
freakishfreakish
12.3k1630
edited Dec 17 '18 at 9:00
answered Dec 16 '18 at 12:24
freakishfreakish
12.3k1630
answered Dec 16 '18 at 12:24
freakishfreakish
12.3k1630
answered Dec 16 '18 at 12:24
freakishfreakish
12.3k1630
12.3k1630
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add a comment |
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