Use of the symbol $G/N$ for the quotient group
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I am reading Dummit and Foote's Abstract Algebra, 3rd edition and have a question about their use of the symbol $G/N$.
On page 76, they define the quotient group as follows:
Definition. Let $varphi:Gto H$ be a homomorphism with kernel $K$. The quotient group, $G/K$, is the group whose elements are the fibers of $varphi$ with the following group operation: if $X$ is the fiber above $a$ and $Y$ is the fiber above $b$ then the product of $X$ with $Y$ is defined to be the fiber above the product $ab$.
So at this point, the quotient group $G/K$ is defined only when we know $K$ is the kernel of some homomorphism.
Later in the section on page 82, they prove the following proposition:
Proposition 7. A subgroup $N$ of the group $G$ is normal if and only if it is the kernel of some homomorphism.
In the proof, they say "if $N$ is a subgroup of $G$, let $H=G/N$". At this point, they haven't proved that every subgroup is the kernel of some homomorphism. In fact, that's what they are trying to prove here. My question is why they can use the notation $G/N$ before proving that $N$ is the kernel?
abstract-algebra proof-explanation
asked Aug 7 at 21:18
kmiyazaki
33711
add a comment |Â
up vote
1
down vote
favorite
I am reading Dummit and Foote's Abstract Algebra, 3rd edition and have a question about their use of the symbol $G/N$.
On page 76, they define the quotient group as follows:
Definition. Let $varphi:Gto H$ be a homomorphism with kernel $K$. The quotient group, $G/K$, is the group whose elements are the fibers of $varphi$ with the following group operation: if $X$ is the fiber above $a$ and $Y$ is the fiber above $b$ then the product of $X$ with $Y$ is defined to be the fiber above the product $ab$.
So at this point, the quotient group $G/K$ is defined only when we know $K$ is the kernel of some homomorphism.
Later in the section on page 82, they prove the following proposition:
Proposition 7. A subgroup $N$ of the group $G$ is normal if and only if it is the kernel of some homomorphism.
In the proof, they say "if $N$ is a subgroup of $G$, let $H=G/N$". At this point, they haven't proved that every subgroup is the kernel of some homomorphism. In fact, that's what they are trying to prove here. My question is why they can use the notation $G/N$ before proving that $N$ is the kernel?
abstract-algebra proof-explanation
asked Aug 7 at 21:18
kmiyazaki
33711
Thank you for correction.
– kmiyazaki
Aug 7 at 21:25
1
I don't have a copy of Dummit and Foote. But check to see if there isn't an alternate definition of a quotient group of $G/N$ as the cosets of $N$ in $G.$
– Doug M
Aug 7 at 21:32
That is what I was hoping to find, but they do not define $G/N$ in any other way.
– kmiyazaki
Aug 8 at 0:33
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am reading Dummit and Foote's Abstract Algebra, 3rd edition and have a question about their use of the symbol $G/N$.
On page 76, they define the quotient group as follows:
Definition. Let $varphi:Gto H$ be a homomorphism with kernel $K$. The quotient group, $G/K$, is the group whose elements are the fibers of $varphi$ with the following group operation: if $X$ is the fiber above $a$ and $Y$ is the fiber above $b$ then the product of $X$ with $Y$ is defined to be the fiber above the product $ab$.
So at this point, the quotient group $G/K$ is defined only when we know $K$ is the kernel of some homomorphism.
Later in the section on page 82, they prove the following proposition:
Proposition 7. A subgroup $N$ of the group $G$ is normal if and only if it is the kernel of some homomorphism.
In the proof, they say "if $N$ is a subgroup of $G$, let $H=G/N$". At this point, they haven't proved that every subgroup is the kernel of some homomorphism. In fact, that's what they are trying to prove here. My question is why they can use the notation $G/N$ before proving that $N$ is the kernel?
abstract-algebra proof-explanation
asked Aug 7 at 21:18
kmiyazaki
33711
I am reading Dummit and Foote's Abstract Algebra, 3rd edition and have a question about their use of the symbol $G/N$.
On page 76, they define the quotient group as follows:
Definition. Let $varphi:Gto H$ be a homomorphism with kernel $K$. The quotient group, $G/K$, is the group whose elements are the fibers of $varphi$ with the following group operation: if $X$ is the fiber above $a$ and $Y$ is the fiber above $b$ then the product of $X$ with $Y$ is defined to be the fiber above the product $ab$.
So at this point, the quotient group $G/K$ is defined only when we know $K$ is the kernel of some homomorphism.
Later in the section on page 82, they prove the following proposition:
Proposition 7. A subgroup $N$ of the group $G$ is normal if and only if it is the kernel of some homomorphism.
In the proof, they say "if $N$ is a subgroup of $G$, let $H=G/N$". At this point, they haven't proved that every subgroup is the kernel of some homomorphism. In fact, that's what they are trying to prove here. My question is why they can use the notation $G/N$ before proving that $N$ is the kernel?
abstract-algebra proof-explanation
asked Aug 7 at 21:18
kmiyazaki
33711
edited Aug 7 at 22:55
asked Aug 7 at 21:18
kmiyazaki
33711
asked Aug 7 at 21:18
kmiyazaki
33711
asked Aug 7 at 21:18
kmiyazaki
33711
33711
Thank you for correction.
– kmiyazaki
Aug 7 at 21:25
1
I don't have a copy of Dummit and Foote. But check to see if there isn't an alternate definition of a quotient group of $G/N$ as the cosets of $N$ in $G.$
– Doug M
Aug 7 at 21:32
That is what I was hoping to find, but they do not define $G/N$ in any other way.
– kmiyazaki
Aug 8 at 0:33
add a comment |Â
Thank you for correction.
– kmiyazaki
Aug 7 at 21:25
1
I don't have a copy of Dummit and Foote. But check to see if there isn't an alternate definition of a quotient group of $G/N$ as the cosets of $N$ in $G.$
– Doug M
Aug 7 at 21:32
That is what I was hoping to find, but they do not define $G/N$ in any other way.
– kmiyazaki
Aug 8 at 0:33
Thank you for correction.
– kmiyazaki
Aug 7 at 21:25
Thank you for correction.
– kmiyazaki
Aug 7 at 21:25
1
1
I don't have a copy of Dummit and Foote. But check to see if there isn't an alternate definition of a quotient group of $G/N$ as the cosets of $N$ in $G.$
– Doug M
Aug 7 at 21:32
I don't have a copy of Dummit and Foote. But check to see if there isn't an alternate definition of a quotient group of $G/N$ as the cosets of $N$ in $G.$
– Doug M
Aug 7 at 21:32
That is what I was hoping to find, but they do not define $G/N$ in any other way.
– kmiyazaki
Aug 8 at 0:33
That is what I was hoping to find, but they do not define $G/N$ in any other way.
– kmiyazaki
Aug 8 at 0:33
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
2
down vote
accepted
Generally speaking, when $H$ is a subgroup of $G$, $G/H$ denotes the set of left cosets of $G$ modulo $H$:
$$G/H=gHmid gin G,$$
and similarly $Hbackslash G$ is the set of right cosets of $G$ modulo $H$:
$$Hbackslash G=Hgmid gin G.$$
If one tries to endow these sets with an operation deduced from the group operation on $G$, one shows this is possible if and only if $H$ is a normal subgroup of $G$. In this case, the left cosets and the right cosets of $G$ modulo $H$ are the same set, which is called the quotient group of $G$ by $H$.
edited Aug 7 at 21:50
egreg
165k1180187
answered Aug 7 at 21:32
Bernard
110k635103
Really? I've never seen $Hsetminus G$ used with that meaning. I've only seen "$setminus$" used to denote exclusion: $Hsetminus G=hin Hmid hnotin G$. Is the notation you mention commonly used?
– MPW
Aug 7 at 21:41
3
It is Bourbaki's notation. Remark the spacing is not the same as with ‘set minus’.
– Bernard
Aug 7 at 21:45
I never would have guessed that. Learned something new today. +1, thanks.
– MPW
Aug 7 at 21:52
@Bernard Thank you for your reply. I am aware of that definition of $G/H$. What I am hoping to understand here is what the authors mean by $G/N$ in their proof given that the only definition of $G/N$ they give is the one using the kernel of a homomorphism.
– kmiyazaki
Aug 8 at 0:42
1
As I don't have this book it's hard to tell precisely. My guess is they have defined this notation for left cosets somewhere before. B.t.w., not every subgroup is the kernel of a homomorphism. Only normal subgroups are.
– Bernard
Aug 8 at 1:26
 |Â
show 1 more comment
up vote
0
down vote
Not sure about your book, but generally the notation $G/H$ is used for the set of left cosets of $H$, that is, $G/H = gH : g in G$. This makes sense for any subgroup $H$. One can try to define a group operation on this set by $(gH)(g'H) = gg'H$, but this is well-defined if and only if $H$ is a normal subgroup. (And once it is, there is a surjective homomorphism $G to G/H$ by $g mapsto gH$, whose kernel is $H$ — this makes it agree with your definition of quotient group.)
answered Aug 7 at 21:31
arkeet
5,005822
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Generally speaking, when $H$ is a subgroup of $G$, $G/H$ denotes the set of left cosets of $G$ modulo $H$:
$$G/H=gHmid gin G,$$
and similarly $Hbackslash G$ is the set of right cosets of $G$ modulo $H$:
$$Hbackslash G=Hgmid gin G.$$
If one tries to endow these sets with an operation deduced from the group operation on $G$, one shows this is possible if and only if $H$ is a normal subgroup of $G$. In this case, the left cosets and the right cosets of $G$ modulo $H$ are the same set, which is called the quotient group of $G$ by $H$.
edited Aug 7 at 21:50
egreg
165k1180187
answered Aug 7 at 21:32
Bernard
110k635103
Really? I've never seen $Hsetminus G$ used with that meaning. I've only seen "$setminus$" used to denote exclusion: $Hsetminus G=hin Hmid hnotin G$. Is the notation you mention commonly used?
– MPW
Aug 7 at 21:41
3
It is Bourbaki's notation. Remark the spacing is not the same as with ‘set minus’.
– Bernard
Aug 7 at 21:45
I never would have guessed that. Learned something new today. +1, thanks.
– MPW
Aug 7 at 21:52
@Bernard Thank you for your reply. I am aware of that definition of $G/H$. What I am hoping to understand here is what the authors mean by $G/N$ in their proof given that the only definition of $G/N$ they give is the one using the kernel of a homomorphism.
– kmiyazaki
Aug 8 at 0:42
1
As I don't have this book it's hard to tell precisely. My guess is they have defined this notation for left cosets somewhere before. B.t.w., not every subgroup is the kernel of a homomorphism. Only normal subgroups are.
– Bernard
Aug 8 at 1:26
 |Â
show 1 more comment
up vote
2
down vote
accepted
Generally speaking, when $H$ is a subgroup of $G$, $G/H$ denotes the set of left cosets of $G$ modulo $H$:
$$G/H=gHmid gin G,$$
and similarly $Hbackslash G$ is the set of right cosets of $G$ modulo $H$:
$$Hbackslash G=Hgmid gin G.$$
If one tries to endow these sets with an operation deduced from the group operation on $G$, one shows this is possible if and only if $H$ is a normal subgroup of $G$. In this case, the left cosets and the right cosets of $G$ modulo $H$ are the same set, which is called the quotient group of $G$ by $H$.
edited Aug 7 at 21:50
egreg
165k1180187
answered Aug 7 at 21:32
Bernard
110k635103
Really? I've never seen $Hsetminus G$ used with that meaning. I've only seen "$setminus$" used to denote exclusion: $Hsetminus G=hin Hmid hnotin G$. Is the notation you mention commonly used?
– MPW
Aug 7 at 21:41
3
It is Bourbaki's notation. Remark the spacing is not the same as with ‘set minus’.
– Bernard
Aug 7 at 21:45
I never would have guessed that. Learned something new today. +1, thanks.
– MPW
Aug 7 at 21:52
@Bernard Thank you for your reply. I am aware of that definition of $G/H$. What I am hoping to understand here is what the authors mean by $G/N$ in their proof given that the only definition of $G/N$ they give is the one using the kernel of a homomorphism.
– kmiyazaki
Aug 8 at 0:42
1
As I don't have this book it's hard to tell precisely. My guess is they have defined this notation for left cosets somewhere before. B.t.w., not every subgroup is the kernel of a homomorphism. Only normal subgroups are.
– Bernard
Aug 8 at 1:26
 |Â
show 1 more comment
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Generally speaking, when $H$ is a subgroup of $G$, $G/H$ denotes the set of left cosets of $G$ modulo $H$:
$$G/H=gHmid gin G,$$
and similarly $Hbackslash G$ is the set of right cosets of $G$ modulo $H$:
$$Hbackslash G=Hgmid gin G.$$
If one tries to endow these sets with an operation deduced from the group operation on $G$, one shows this is possible if and only if $H$ is a normal subgroup of $G$. In this case, the left cosets and the right cosets of $G$ modulo $H$ are the same set, which is called the quotient group of $G$ by $H$.
edited Aug 7 at 21:50
egreg
165k1180187
answered Aug 7 at 21:32
Bernard
110k635103
Generally speaking, when $H$ is a subgroup of $G$, $G/H$ denotes the set of left cosets of $G$ modulo $H$:
$$G/H=gHmid gin G,$$
and similarly $Hbackslash G$ is the set of right cosets of $G$ modulo $H$:
$$Hbackslash G=Hgmid gin G.$$
If one tries to endow these sets with an operation deduced from the group operation on $G$, one shows this is possible if and only if $H$ is a normal subgroup of $G$. In this case, the left cosets and the right cosets of $G$ modulo $H$ are the same set, which is called the quotient group of $G$ by $H$.
edited Aug 7 at 21:50
egreg
165k1180187
answered Aug 7 at 21:32
Bernard
110k635103
edited Aug 7 at 21:50
egreg
165k1180187
edited Aug 7 at 21:50
egreg
165k1180187
edited Aug 7 at 21:50
egreg
165k1180187
165k1180187
answered Aug 7 at 21:32
Bernard
110k635103
answered Aug 7 at 21:32
Bernard
110k635103
answered Aug 7 at 21:32
Bernard
110k635103
110k635103
Really? I've never seen $Hsetminus G$ used with that meaning. I've only seen "$setminus$" used to denote exclusion: $Hsetminus G=hin Hmid hnotin G$. Is the notation you mention commonly used?
– MPW
Aug 7 at 21:41
3
It is Bourbaki's notation. Remark the spacing is not the same as with ‘set minus’.
– Bernard
Aug 7 at 21:45
I never would have guessed that. Learned something new today. +1, thanks.
– MPW
Aug 7 at 21:52
@Bernard Thank you for your reply. I am aware of that definition of $G/H$. What I am hoping to understand here is what the authors mean by $G/N$ in their proof given that the only definition of $G/N$ they give is the one using the kernel of a homomorphism.
– kmiyazaki
Aug 8 at 0:42
1
As I don't have this book it's hard to tell precisely. My guess is they have defined this notation for left cosets somewhere before. B.t.w., not every subgroup is the kernel of a homomorphism. Only normal subgroups are.
– Bernard
Aug 8 at 1:26
 |Â
show 1 more comment
Really? I've never seen $Hsetminus G$ used with that meaning. I've only seen "$setminus$" used to denote exclusion: $Hsetminus G=hin Hmid hnotin G$. Is the notation you mention commonly used?
– MPW
Aug 7 at 21:41
3
It is Bourbaki's notation. Remark the spacing is not the same as with ‘set minus’.
– Bernard
Aug 7 at 21:45
I never would have guessed that. Learned something new today. +1, thanks.
– MPW
Aug 7 at 21:52
@Bernard Thank you for your reply. I am aware of that definition of $G/H$. What I am hoping to understand here is what the authors mean by $G/N$ in their proof given that the only definition of $G/N$ they give is the one using the kernel of a homomorphism.
– kmiyazaki
Aug 8 at 0:42
1
As I don't have this book it's hard to tell precisely. My guess is they have defined this notation for left cosets somewhere before. B.t.w., not every subgroup is the kernel of a homomorphism. Only normal subgroups are.
– Bernard
Aug 8 at 1:26
Really? I've never seen $Hsetminus G$ used with that meaning. I've only seen "$setminus$" used to denote exclusion: $Hsetminus G=hin Hmid hnotin G$. Is the notation you mention commonly used?
– MPW
Aug 7 at 21:41
Really? I've never seen $Hsetminus G$ used with that meaning. I've only seen "$setminus$" used to denote exclusion: $Hsetminus G=hin Hmid hnotin G$. Is the notation you mention commonly used?
– MPW
Aug 7 at 21:41
3
3
It is Bourbaki's notation. Remark the spacing is not the same as with ‘set minus’.
– Bernard
Aug 7 at 21:45
It is Bourbaki's notation. Remark the spacing is not the same as with ‘set minus’.
– Bernard
Aug 7 at 21:45
I never would have guessed that. Learned something new today. +1, thanks.
– MPW
Aug 7 at 21:52
I never would have guessed that. Learned something new today. +1, thanks.
– MPW
Aug 7 at 21:52
@Bernard Thank you for your reply. I am aware of that definition of $G/H$. What I am hoping to understand here is what the authors mean by $G/N$ in their proof given that the only definition of $G/N$ they give is the one using the kernel of a homomorphism.
– kmiyazaki
Aug 8 at 0:42
@Bernard Thank you for your reply. I am aware of that definition of $G/H$. What I am hoping to understand here is what the authors mean by $G/N$ in their proof given that the only definition of $G/N$ they give is the one using the kernel of a homomorphism.
– kmiyazaki
Aug 8 at 0:42
1
1
As I don't have this book it's hard to tell precisely. My guess is they have defined this notation for left cosets somewhere before. B.t.w., not every subgroup is the kernel of a homomorphism. Only normal subgroups are.
– Bernard
Aug 8 at 1:26
As I don't have this book it's hard to tell precisely. My guess is they have defined this notation for left cosets somewhere before. B.t.w., not every subgroup is the kernel of a homomorphism. Only normal subgroups are.
– Bernard
Aug 8 at 1:26
 |Â
show 1 more comment
up vote
0
down vote
Not sure about your book, but generally the notation $G/H$ is used for the set of left cosets of $H$, that is, $G/H = gH : g in G$. This makes sense for any subgroup $H$. One can try to define a group operation on this set by $(gH)(g'H) = gg'H$, but this is well-defined if and only if $H$ is a normal subgroup. (And once it is, there is a surjective homomorphism $G to G/H$ by $g mapsto gH$, whose kernel is $H$ — this makes it agree with your definition of quotient group.)
answered Aug 7 at 21:31
arkeet
5,005822
add a comment |Â
up vote
0
down vote
Not sure about your book, but generally the notation $G/H$ is used for the set of left cosets of $H$, that is, $G/H = gH : g in G$. This makes sense for any subgroup $H$. One can try to define a group operation on this set by $(gH)(g'H) = gg'H$, but this is well-defined if and only if $H$ is a normal subgroup. (And once it is, there is a surjective homomorphism $G to G/H$ by $g mapsto gH$, whose kernel is $H$ — this makes it agree with your definition of quotient group.)
answered Aug 7 at 21:31
arkeet
5,005822
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Not sure about your book, but generally the notation $G/H$ is used for the set of left cosets of $H$, that is, $G/H = gH : g in G$. This makes sense for any subgroup $H$. One can try to define a group operation on this set by $(gH)(g'H) = gg'H$, but this is well-defined if and only if $H$ is a normal subgroup. (And once it is, there is a surjective homomorphism $G to G/H$ by $g mapsto gH$, whose kernel is $H$ — this makes it agree with your definition of quotient group.)
answered Aug 7 at 21:31
arkeet
5,005822
Not sure about your book, but generally the notation $G/H$ is used for the set of left cosets of $H$, that is, $G/H = gH : g in G$. This makes sense for any subgroup $H$. One can try to define a group operation on this set by $(gH)(g'H) = gg'H$, but this is well-defined if and only if $H$ is a normal subgroup. (And once it is, there is a surjective homomorphism $G to G/H$ by $g mapsto gH$, whose kernel is $H$ — this makes it agree with your definition of quotient group.)
answered Aug 7 at 21:31
arkeet
5,005822
answered Aug 7 at 21:31
arkeet
5,005822
answered Aug 7 at 21:31
arkeet
5,005822
answered Aug 7 at 21:31
arkeet
5,005822
5,005822
add a comment |Â
add a comment |Â
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Thank you for correction.
– kmiyazaki
Aug 7 at 21:25
1
I don't have a copy of Dummit and Foote. But check to see if there isn't an alternate definition of a quotient group of $G/N$ as the cosets of $N$ in $G.$
– Doug M
Aug 7 at 21:32
That is what I was hoping to find, but they do not define $G/N$ in any other way.
– kmiyazaki
Aug 8 at 0:33