Almost normal subgroups: Is there any notion which is weaker than normal subgroup?
$begingroup$
Let $G$ be a group then $N$ a subgroup of $G$ is said to be normal subgroup of $G$ if $forall g in G$, $g^{-1}Ng = N$.
Is there any notion which is weaker than normal subgroup? I mean something like almost normal subgroup or nearly normal subgroup in which some of the elements only follow the normality condition.
group-theory definition normal-subgroups
edited Feb 21 at 15:50
Shaun
9,310113684
asked Feb 21 at 6:10
I_wil_break_wallI_wil_break_wall
805
$endgroup$
add a comment |
$begingroup$
Let $G$ be a group then $N$ a subgroup of $G$ is said to be normal subgroup of $G$ if $forall g in G$, $g^{-1}Ng = N$.
Is there any notion which is weaker than normal subgroup? I mean something like almost normal subgroup or nearly normal subgroup in which some of the elements only follow the normality condition.
group-theory definition normal-subgroups
edited Feb 21 at 15:50
Shaun
9,310113684
asked Feb 21 at 6:10
I_wil_break_wallI_wil_break_wall
805
$endgroup$
3
$begingroup$
Groupprops lists some - groupprops.subwiki.org/wiki/Normal_subgroup#Weaker_properties
$endgroup$
– Eevee Trainer
Feb 21 at 6:11
add a comment |
$begingroup$
Let $G$ be a group then $N$ a subgroup of $G$ is said to be normal subgroup of $G$ if $forall g in G$, $g^{-1}Ng = N$.
Is there any notion which is weaker than normal subgroup? I mean something like almost normal subgroup or nearly normal subgroup in which some of the elements only follow the normality condition.
group-theory definition normal-subgroups
edited Feb 21 at 15:50
Shaun
9,310113684
asked Feb 21 at 6:10
I_wil_break_wallI_wil_break_wall
805
$endgroup$
Let $G$ be a group then $N$ a subgroup of $G$ is said to be normal subgroup of $G$ if $forall g in G$, $g^{-1}Ng = N$.
Is there any notion which is weaker than normal subgroup? I mean something like almost normal subgroup or nearly normal subgroup in which some of the elements only follow the normality condition.
group-theory definition normal-subgroups
group-theory definition normal-subgroups
edited Feb 21 at 15:50
Shaun
9,310113684
asked Feb 21 at 6:10
I_wil_break_wallI_wil_break_wall
805
edited Feb 21 at 15:50
Shaun
9,310113684
asked Feb 21 at 6:10
I_wil_break_wallI_wil_break_wall
805
edited Feb 21 at 15:50
Shaun
9,310113684
edited Feb 21 at 15:50
Shaun
9,310113684
edited Feb 21 at 15:50
Shaun
9,310113684
9,310113684
asked Feb 21 at 6:10
I_wil_break_wallI_wil_break_wall
805
asked Feb 21 at 6:10
I_wil_break_wallI_wil_break_wall
805
asked Feb 21 at 6:10
I_wil_break_wallI_wil_break_wall
805
805
3
$begingroup$
Groupprops lists some - groupprops.subwiki.org/wiki/Normal_subgroup#Weaker_properties
$endgroup$
– Eevee Trainer
Feb 21 at 6:11
add a comment |
3
$begingroup$
Groupprops lists some - groupprops.subwiki.org/wiki/Normal_subgroup#Weaker_properties
$endgroup$
– Eevee Trainer
Feb 21 at 6:11
3
3
$begingroup$
Groupprops lists some - groupprops.subwiki.org/wiki/Normal_subgroup#Weaker_properties
$endgroup$
– Eevee Trainer
Feb 21 at 6:11
$begingroup$
Groupprops lists some - groupprops.subwiki.org/wiki/Normal_subgroup#Weaker_properties
$endgroup$
– Eevee Trainer
Feb 21 at 6:11
add a comment |
5 Answers
5
active
oldest
votes
$begingroup$
A very important notion is that of subnormality: a subgroup $H$ of $G$ is called subnormal if there is a series $H=H_0 lhd H_1 lhd cdots lhd H_{n-1} lhd H_n=G$. Of course every normal subgroup is subnormal, but the converse is not true in general. Helmut Wielandt basically established the theory in 1939, one of his famous results being the Zipper Lemma.
Other notions to be considered are pronormal subgroups and quasinormal or permutable subgroups, each of which generalizes a certain property of normal subgroups.
answered Feb 21 at 9:54
Nicky HeksterNicky Hekster
28.8k63456
$endgroup$
add a comment |
$begingroup$
Consider the set of all conjugates of a subgroup $Hleq G$, defined by $C={gHg^{-1}mid gin G}$. If $H$ is normal, clearly $|C|=1$, so $|C|$ can be considered to be a measure of "how far away from being normal" a subgroup is.
answered Feb 21 at 7:19
YiFanYiFan
4,3391627
$endgroup$
add a comment |
$begingroup$
What about $N$ admits only a finite number oof conjugates ? For instance, when the group $G$ acts on a finite space, the stabilizer of a point has this property.
answered Feb 21 at 6:59
ThomasThomas
4,102510
$endgroup$
$begingroup$
The second sentence characterizes finite index subgroups.
$endgroup$
– YCor
Feb 22 at 10:14
add a comment |
$begingroup$
In the contex of locally compact topological groups which are equiped with the Haar measure, one can consider the following:
"For almost all $gin G$ we have $g^{-1} Ng=N$"
In the context of Lie groups, one can consider $g^{-1} N g$ would be isometric to $N$ with metric they inherit from a left (but not right) invariant metric.
In the context of abstract groups one can says "$cap_g g^{-1} N g$ has finite index in $N$.
answered Feb 21 at 7:37
Ali TaghaviAli Taghavi
257330
$endgroup$
add a comment |
$begingroup$
$N$ is said to be a commensurated subgroup if $[N:Ncap gNg^{-1}]$ is finite for every $gin G$. This has many other names (almost normal, $(G,N)$ is a Hecke pair, etc). This notion occurs in various contexts.
answered Feb 22 at 10:14
YCorYCor
7,798929
$endgroup$
add a comment |
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5 Answers
5
active
oldest
votes
5 Answers
5
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
A very important notion is that of subnormality: a subgroup $H$ of $G$ is called subnormal if there is a series $H=H_0 lhd H_1 lhd cdots lhd H_{n-1} lhd H_n=G$. Of course every normal subgroup is subnormal, but the converse is not true in general. Helmut Wielandt basically established the theory in 1939, one of his famous results being the Zipper Lemma.
Other notions to be considered are pronormal subgroups and quasinormal or permutable subgroups, each of which generalizes a certain property of normal subgroups.
answered Feb 21 at 9:54
Nicky HeksterNicky Hekster
28.8k63456
$endgroup$
add a comment |
$begingroup$
A very important notion is that of subnormality: a subgroup $H$ of $G$ is called subnormal if there is a series $H=H_0 lhd H_1 lhd cdots lhd H_{n-1} lhd H_n=G$. Of course every normal subgroup is subnormal, but the converse is not true in general. Helmut Wielandt basically established the theory in 1939, one of his famous results being the Zipper Lemma.
Other notions to be considered are pronormal subgroups and quasinormal or permutable subgroups, each of which generalizes a certain property of normal subgroups.
answered Feb 21 at 9:54
Nicky HeksterNicky Hekster
28.8k63456
$endgroup$
add a comment |
$begingroup$
A very important notion is that of subnormality: a subgroup $H$ of $G$ is called subnormal if there is a series $H=H_0 lhd H_1 lhd cdots lhd H_{n-1} lhd H_n=G$. Of course every normal subgroup is subnormal, but the converse is not true in general. Helmut Wielandt basically established the theory in 1939, one of his famous results being the Zipper Lemma.
Other notions to be considered are pronormal subgroups and quasinormal or permutable subgroups, each of which generalizes a certain property of normal subgroups.
answered Feb 21 at 9:54
Nicky HeksterNicky Hekster
28.8k63456
$endgroup$
A very important notion is that of subnormality: a subgroup $H$ of $G$ is called subnormal if there is a series $H=H_0 lhd H_1 lhd cdots lhd H_{n-1} lhd H_n=G$. Of course every normal subgroup is subnormal, but the converse is not true in general. Helmut Wielandt basically established the theory in 1939, one of his famous results being the Zipper Lemma.
Other notions to be considered are pronormal subgroups and quasinormal or permutable subgroups, each of which generalizes a certain property of normal subgroups.
answered Feb 21 at 9:54
Nicky HeksterNicky Hekster
28.8k63456
edited Feb 21 at 16:10
answered Feb 21 at 9:54
Nicky HeksterNicky Hekster
28.8k63456
answered Feb 21 at 9:54
Nicky HeksterNicky Hekster
28.8k63456
answered Feb 21 at 9:54
Nicky HeksterNicky Hekster
28.8k63456
28.8k63456
add a comment |
add a comment |
$begingroup$
Consider the set of all conjugates of a subgroup $Hleq G$, defined by $C={gHg^{-1}mid gin G}$. If $H$ is normal, clearly $|C|=1$, so $|C|$ can be considered to be a measure of "how far away from being normal" a subgroup is.
answered Feb 21 at 7:19
YiFanYiFan
4,3391627
$endgroup$
add a comment |
$begingroup$
Consider the set of all conjugates of a subgroup $Hleq G$, defined by $C={gHg^{-1}mid gin G}$. If $H$ is normal, clearly $|C|=1$, so $|C|$ can be considered to be a measure of "how far away from being normal" a subgroup is.
answered Feb 21 at 7:19
YiFanYiFan
4,3391627
$endgroup$
add a comment |
$begingroup$
Consider the set of all conjugates of a subgroup $Hleq G$, defined by $C={gHg^{-1}mid gin G}$. If $H$ is normal, clearly $|C|=1$, so $|C|$ can be considered to be a measure of "how far away from being normal" a subgroup is.
answered Feb 21 at 7:19
YiFanYiFan
4,3391627
$endgroup$
Consider the set of all conjugates of a subgroup $Hleq G$, defined by $C={gHg^{-1}mid gin G}$. If $H$ is normal, clearly $|C|=1$, so $|C|$ can be considered to be a measure of "how far away from being normal" a subgroup is.
answered Feb 21 at 7:19
YiFanYiFan
4,3391627
answered Feb 21 at 7:19
YiFanYiFan
4,3391627
answered Feb 21 at 7:19
YiFanYiFan
4,3391627
answered Feb 21 at 7:19
YiFanYiFan
4,3391627
4,3391627
add a comment |
add a comment |
$begingroup$
What about $N$ admits only a finite number oof conjugates ? For instance, when the group $G$ acts on a finite space, the stabilizer of a point has this property.
answered Feb 21 at 6:59
ThomasThomas
4,102510
$endgroup$
$begingroup$
The second sentence characterizes finite index subgroups.
$endgroup$
– YCor
Feb 22 at 10:14
add a comment |
$begingroup$
What about $N$ admits only a finite number oof conjugates ? For instance, when the group $G$ acts on a finite space, the stabilizer of a point has this property.
answered Feb 21 at 6:59
ThomasThomas
4,102510
$endgroup$
$begingroup$
The second sentence characterizes finite index subgroups.
$endgroup$
– YCor
Feb 22 at 10:14
add a comment |
$begingroup$
What about $N$ admits only a finite number oof conjugates ? For instance, when the group $G$ acts on a finite space, the stabilizer of a point has this property.
answered Feb 21 at 6:59
ThomasThomas
4,102510
$endgroup$
What about $N$ admits only a finite number oof conjugates ? For instance, when the group $G$ acts on a finite space, the stabilizer of a point has this property.
answered Feb 21 at 6:59
ThomasThomas
4,102510
answered Feb 21 at 6:59
ThomasThomas
4,102510
answered Feb 21 at 6:59
ThomasThomas
4,102510
answered Feb 21 at 6:59
ThomasThomas
4,102510
4,102510
$begingroup$
The second sentence characterizes finite index subgroups.
$endgroup$
– YCor
Feb 22 at 10:14
add a comment |
$begingroup$
The second sentence characterizes finite index subgroups.
$endgroup$
– YCor
Feb 22 at 10:14
$begingroup$
The second sentence characterizes finite index subgroups.
$endgroup$
– YCor
Feb 22 at 10:14
$begingroup$
The second sentence characterizes finite index subgroups.
$endgroup$
– YCor
Feb 22 at 10:14
add a comment |
$begingroup$
In the contex of locally compact topological groups which are equiped with the Haar measure, one can consider the following:
"For almost all $gin G$ we have $g^{-1} Ng=N$"
In the context of Lie groups, one can consider $g^{-1} N g$ would be isometric to $N$ with metric they inherit from a left (but not right) invariant metric.
In the context of abstract groups one can says "$cap_g g^{-1} N g$ has finite index in $N$.
answered Feb 21 at 7:37
Ali TaghaviAli Taghavi
257330
$endgroup$
add a comment |
$begingroup$
In the contex of locally compact topological groups which are equiped with the Haar measure, one can consider the following:
"For almost all $gin G$ we have $g^{-1} Ng=N$"
In the context of Lie groups, one can consider $g^{-1} N g$ would be isometric to $N$ with metric they inherit from a left (but not right) invariant metric.
In the context of abstract groups one can says "$cap_g g^{-1} N g$ has finite index in $N$.
answered Feb 21 at 7:37
Ali TaghaviAli Taghavi
257330
$endgroup$
add a comment |
$begingroup$
In the contex of locally compact topological groups which are equiped with the Haar measure, one can consider the following:
"For almost all $gin G$ we have $g^{-1} Ng=N$"
In the context of Lie groups, one can consider $g^{-1} N g$ would be isometric to $N$ with metric they inherit from a left (but not right) invariant metric.
In the context of abstract groups one can says "$cap_g g^{-1} N g$ has finite index in $N$.
answered Feb 21 at 7:37
Ali TaghaviAli Taghavi
257330
$endgroup$
In the contex of locally compact topological groups which are equiped with the Haar measure, one can consider the following:
"For almost all $gin G$ we have $g^{-1} Ng=N$"
In the context of Lie groups, one can consider $g^{-1} N g$ would be isometric to $N$ with metric they inherit from a left (but not right) invariant metric.
In the context of abstract groups one can says "$cap_g g^{-1} N g$ has finite index in $N$.
answered Feb 21 at 7:37
Ali TaghaviAli Taghavi
257330
edited Feb 21 at 9:30
answered Feb 21 at 7:37
Ali TaghaviAli Taghavi
257330
answered Feb 21 at 7:37
Ali TaghaviAli Taghavi
257330
answered Feb 21 at 7:37
Ali TaghaviAli Taghavi
257330
257330
add a comment |
add a comment |
$begingroup$
$N$ is said to be a commensurated subgroup if $[N:Ncap gNg^{-1}]$ is finite for every $gin G$. This has many other names (almost normal, $(G,N)$ is a Hecke pair, etc). This notion occurs in various contexts.
answered Feb 22 at 10:14
YCorYCor
7,798929
$endgroup$
add a comment |
$begingroup$
$N$ is said to be a commensurated subgroup if $[N:Ncap gNg^{-1}]$ is finite for every $gin G$. This has many other names (almost normal, $(G,N)$ is a Hecke pair, etc). This notion occurs in various contexts.
answered Feb 22 at 10:14
YCorYCor
7,798929
$endgroup$
add a comment |
$begingroup$
$N$ is said to be a commensurated subgroup if $[N:Ncap gNg^{-1}]$ is finite for every $gin G$. This has many other names (almost normal, $(G,N)$ is a Hecke pair, etc). This notion occurs in various contexts.
answered Feb 22 at 10:14
YCorYCor
7,798929
$endgroup$
$N$ is said to be a commensurated subgroup if $[N:Ncap gNg^{-1}]$ is finite for every $gin G$. This has many other names (almost normal, $(G,N)$ is a Hecke pair, etc). This notion occurs in various contexts.
answered Feb 22 at 10:14
YCorYCor
7,798929
answered Feb 22 at 10:14
YCorYCor
7,798929
answered Feb 22 at 10:14
YCorYCor
7,798929
answered Feb 22 at 10:14
YCorYCor
7,798929
7,798929
add a comment |
add a comment |
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3
$begingroup$
Groupprops lists some - groupprops.subwiki.org/wiki/Normal_subgroup#Weaker_properties
$endgroup$
– Eevee Trainer
Feb 21 at 6:11