Mixing
When does a simple Lie group contain a nonsimple subgroup?
$begingroup$
The group of rotations of Euclidean space in $N$ dimensions is the special orthogonal group $text{SO}(N)$. It is simple and all its Lie subgroups are (semi)simple as well.
The conformal group of Euclidean space is simple too; it's the indefinite special orthogonal group $text{SO}(N+1,1)$. But one of its subgroups is the Euclidean isometry group, also called the inhomogeneous special orthogonal group $text{ISO}(N)$. This subgroup is not simple: $text{ISO}(N) = mathbb R^N rtimes text{SO}(N)$.
The above was just an example. In general, is there any way I can know if a simple group will or will not have nonsimple subgroups?
group-theory lie-groups simple-groups
asked Jan 22 at 10:16
TotofofoTotofofo
161
$endgroup$
add a comment |
$begingroup$
The group of rotations of Euclidean space in $N$ dimensions is the special orthogonal group $text{SO}(N)$. It is simple and all its Lie subgroups are (semi)simple as well.
The conformal group of Euclidean space is simple too; it's the indefinite special orthogonal group $text{SO}(N+1,1)$. But one of its subgroups is the Euclidean isometry group, also called the inhomogeneous special orthogonal group $text{ISO}(N)$. This subgroup is not simple: $text{ISO}(N) = mathbb R^N rtimes text{SO}(N)$.
The above was just an example. In general, is there any way I can know if a simple group will or will not have nonsimple subgroups?
group-theory lie-groups simple-groups
asked Jan 22 at 10:16
TotofofoTotofofo
161
$endgroup$
$begingroup$
I expect this should happen a lot; should it only be for Lie groups of dimension $0$ , of which finite groups are special cases (try $mathfrak{A}_n$, it contains many nonsimple subgroups). I don't know if there's an easy criterion though. Any simple Lie group that contains a free group will do
$endgroup$
– Max
Jan 22 at 10:54
1
$begingroup$
Your claim "all Lie subgroups of $SO(N)$ are semisimple" is false (for $Nge 2$). Every nontrivial connected Lie group has closed connected non-simple (abelian) subgroups.
$endgroup$
– YCor
Jan 22 at 18:40
add a comment |
$begingroup$
The group of rotations of Euclidean space in $N$ dimensions is the special orthogonal group $text{SO}(N)$. It is simple and all its Lie subgroups are (semi)simple as well.
The conformal group of Euclidean space is simple too; it's the indefinite special orthogonal group $text{SO}(N+1,1)$. But one of its subgroups is the Euclidean isometry group, also called the inhomogeneous special orthogonal group $text{ISO}(N)$. This subgroup is not simple: $text{ISO}(N) = mathbb R^N rtimes text{SO}(N)$.
The above was just an example. In general, is there any way I can know if a simple group will or will not have nonsimple subgroups?
group-theory lie-groups simple-groups
asked Jan 22 at 10:16
TotofofoTotofofo
161
$endgroup$
The group of rotations of Euclidean space in $N$ dimensions is the special orthogonal group $text{SO}(N)$. It is simple and all its Lie subgroups are (semi)simple as well.
The conformal group of Euclidean space is simple too; it's the indefinite special orthogonal group $text{SO}(N+1,1)$. But one of its subgroups is the Euclidean isometry group, also called the inhomogeneous special orthogonal group $text{ISO}(N)$. This subgroup is not simple: $text{ISO}(N) = mathbb R^N rtimes text{SO}(N)$.
The above was just an example. In general, is there any way I can know if a simple group will or will not have nonsimple subgroups?
group-theory lie-groups simple-groups
group-theory lie-groups simple-groups
asked Jan 22 at 10:16
TotofofoTotofofo
161
asked Jan 22 at 10:16
TotofofoTotofofo
161
asked Jan 22 at 10:16
TotofofoTotofofo
161
asked Jan 22 at 10:16
TotofofoTotofofo
161
asked Jan 22 at 10:16
TotofofoTotofofo
161
161
$begingroup$
I expect this should happen a lot; should it only be for Lie groups of dimension $0$ , of which finite groups are special cases (try $mathfrak{A}_n$, it contains many nonsimple subgroups). I don't know if there's an easy criterion though. Any simple Lie group that contains a free group will do
$endgroup$
– Max
Jan 22 at 10:54
1
$begingroup$
Your claim "all Lie subgroups of $SO(N)$ are semisimple" is false (for $Nge 2$). Every nontrivial connected Lie group has closed connected non-simple (abelian) subgroups.
$endgroup$
– YCor
Jan 22 at 18:40
add a comment |
$begingroup$
I expect this should happen a lot; should it only be for Lie groups of dimension $0$ , of which finite groups are special cases (try $mathfrak{A}_n$, it contains many nonsimple subgroups). I don't know if there's an easy criterion though. Any simple Lie group that contains a free group will do
$endgroup$
– Max
Jan 22 at 10:54
1
$begingroup$
Your claim "all Lie subgroups of $SO(N)$ are semisimple" is false (for $Nge 2$). Every nontrivial connected Lie group has closed connected non-simple (abelian) subgroups.
$endgroup$
– YCor
Jan 22 at 18:40
$begingroup$
I expect this should happen a lot; should it only be for Lie groups of dimension $0$ , of which finite groups are special cases (try $mathfrak{A}_n$, it contains many nonsimple subgroups). I don't know if there's an easy criterion though. Any simple Lie group that contains a free group will do
$endgroup$
– Max
Jan 22 at 10:54
$begingroup$
I expect this should happen a lot; should it only be for Lie groups of dimension $0$ , of which finite groups are special cases (try $mathfrak{A}_n$, it contains many nonsimple subgroups). I don't know if there's an easy criterion though. Any simple Lie group that contains a free group will do
$endgroup$
– Max
Jan 22 at 10:54
1
1
$begingroup$
Your claim "all Lie subgroups of $SO(N)$ are semisimple" is false (for $Nge 2$). Every nontrivial connected Lie group has closed connected non-simple (abelian) subgroups.
$endgroup$
– YCor
Jan 22 at 18:40
$begingroup$
Your claim "all Lie subgroups of $SO(N)$ are semisimple" is false (for $Nge 2$). Every nontrivial connected Lie group has closed connected non-simple (abelian) subgroups.
$endgroup$
– YCor
Jan 22 at 18:40
add a comment |
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$begingroup$
I expect this should happen a lot; should it only be for Lie groups of dimension $0$ , of which finite groups are special cases (try $mathfrak{A}_n$, it contains many nonsimple subgroups). I don't know if there's an easy criterion though. Any simple Lie group that contains a free group will do
$endgroup$
– Max
Jan 22 at 10:54
1
$begingroup$
Your claim "all Lie subgroups of $SO(N)$ are semisimple" is false (for $Nge 2$). Every nontrivial connected Lie group has closed connected non-simple (abelian) subgroups.
$endgroup$
– YCor
Jan 22 at 18:40