Cyclic notation for residue classes mod $n$
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I am reading an example from Dummit and Foote's Abstract Algebra, 3rd edition.
I am wondering why the authors do not use the bar notation for the element of residue classes.
For example, why do they write $mathbbZ/2mathbbZ=langle~1~rangle$ instead of $mathbbZ/2mathbbZ=langle~bar1~rangle$?
The reason why I am confused is that they once use the bar notation when they introduce groups $mathbbZ/nmathbbZ$ and suddenly switche to this notation. Is it a common practice not to use the bar when denoting an element of $mathbbZ/nmathbbZ$, or is there any other reason for why they do not use the bar notation here?
abstract-algebra group-theory notation cyclic-groups
asked Aug 17 at 1:10
kmiyazaki
33711
add a comment |Â
up vote
1
down vote
favorite
I am reading an example from Dummit and Foote's Abstract Algebra, 3rd edition.
I am wondering why the authors do not use the bar notation for the element of residue classes.
For example, why do they write $mathbbZ/2mathbbZ=langle~1~rangle$ instead of $mathbbZ/2mathbbZ=langle~bar1~rangle$?
The reason why I am confused is that they once use the bar notation when they introduce groups $mathbbZ/nmathbbZ$ and suddenly switche to this notation. Is it a common practice not to use the bar when denoting an element of $mathbbZ/nmathbbZ$, or is there any other reason for why they do not use the bar notation here?
abstract-algebra group-theory notation cyclic-groups
asked Aug 17 at 1:10
kmiyazaki
33711
2
It's very common practice. Also, it is aligned with defining $n:=overbrace1+dots+1^n$ where $1$ is the multiplicative identity.
– Berci
Aug 17 at 1:22
2
Try typsetting a paper or 200-page thesis with bars everywhere. You'll come around.
– Randall
Aug 17 at 1:26
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am reading an example from Dummit and Foote's Abstract Algebra, 3rd edition.
I am wondering why the authors do not use the bar notation for the element of residue classes.
For example, why do they write $mathbbZ/2mathbbZ=langle~1~rangle$ instead of $mathbbZ/2mathbbZ=langle~bar1~rangle$?
The reason why I am confused is that they once use the bar notation when they introduce groups $mathbbZ/nmathbbZ$ and suddenly switche to this notation. Is it a common practice not to use the bar when denoting an element of $mathbbZ/nmathbbZ$, or is there any other reason for why they do not use the bar notation here?
abstract-algebra group-theory notation cyclic-groups
asked Aug 17 at 1:10
kmiyazaki
33711
I am reading an example from Dummit and Foote's Abstract Algebra, 3rd edition.
I am wondering why the authors do not use the bar notation for the element of residue classes.
For example, why do they write $mathbbZ/2mathbbZ=langle~1~rangle$ instead of $mathbbZ/2mathbbZ=langle~bar1~rangle$?
The reason why I am confused is that they once use the bar notation when they introduce groups $mathbbZ/nmathbbZ$ and suddenly switche to this notation. Is it a common practice not to use the bar when denoting an element of $mathbbZ/nmathbbZ$, or is there any other reason for why they do not use the bar notation here?
abstract-algebra group-theory notation cyclic-groups
asked Aug 17 at 1:10
kmiyazaki
33711
asked Aug 17 at 1:10
kmiyazaki
33711
asked Aug 17 at 1:10
kmiyazaki
33711
asked Aug 17 at 1:10
kmiyazaki
33711
33711
2
It's very common practice. Also, it is aligned with defining $n:=overbrace1+dots+1^n$ where $1$ is the multiplicative identity.
– Berci
Aug 17 at 1:22
2
Try typsetting a paper or 200-page thesis with bars everywhere. You'll come around.
– Randall
Aug 17 at 1:26
add a comment |Â
2
It's very common practice. Also, it is aligned with defining $n:=overbrace1+dots+1^n$ where $1$ is the multiplicative identity.
– Berci
Aug 17 at 1:22
2
Try typsetting a paper or 200-page thesis with bars everywhere. You'll come around.
– Randall
Aug 17 at 1:26
2
2
It's very common practice. Also, it is aligned with defining $n:=overbrace1+dots+1^n$ where $1$ is the multiplicative identity.
– Berci
Aug 17 at 1:22
It's very common practice. Also, it is aligned with defining $n:=overbrace1+dots+1^n$ where $1$ is the multiplicative identity.
– Berci
Aug 17 at 1:22
2
2
Try typsetting a paper or 200-page thesis with bars everywhere. You'll come around.
– Randall
Aug 17 at 1:26
Try typsetting a paper or 200-page thesis with bars everywhere. You'll come around.
– Randall
Aug 17 at 1:26
add a comment |Â
1 Answer
1
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oldest
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up vote
4
down vote
accepted
As you can see the author is now discussing the subgroups of $G=Z_n$, not $BbbZ/nBbbZ$, that's why different notation is used.
In the book, the author denotes $Z_n=0,1,dots,n-1$ and $BbbZ/nBbbZ=bar0,bar1,dots,overlinen-1$
answered Aug 17 at 1:47
Alan Wang
4,486932
I try to avoid using $mathbb Z_n$ because it causes so much confusion, but if I do use it then I define it to be equal to $mathbb Z/nmathbb Z$. Denoting $bar1$ by $1$ is just a convenient abuse of notation.
– Derek Holt
Aug 17 at 7:39
@DerekHolt Ya actually both notation are the same in the sense that the two groups stated are isomorphic, and denoting $bar1$ by $1$ is simpler.
– Alan Wang
Aug 17 at 9:26
@DerekHolt: Do you think calling the multiplicative identity in a (unitary) ring $1$ is always an "abuse of notation", or is that only when the ring happens to be a quotient of $mathbb Z$?
– Henning Makholm
Aug 17 at 9:27
@HenningMakholm I think perhaps that abuse of notation was not exactly what I meant. I meant a simplification of notation to be used when there is no danger of confusion. So I would denote the multiplicative identity in a ring $R$ by $1_R$, but say we will usually just write $1$. Unfortunately many students do succeed in getting confused!
– Derek Holt
Aug 17 at 10:27
@AlanWang I thought that even if it was a convention to denote the elements of $mathbbZ/nmathbbZ$ simply by $0, 1, ldots, n-1$, the authors would say so before using the convention. And it was actually the case. They mention it on page 10. Thank you.
– kmiyazaki
Aug 17 at 19:32
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
As you can see the author is now discussing the subgroups of $G=Z_n$, not $BbbZ/nBbbZ$, that's why different notation is used.
In the book, the author denotes $Z_n=0,1,dots,n-1$ and $BbbZ/nBbbZ=bar0,bar1,dots,overlinen-1$
answered Aug 17 at 1:47
Alan Wang
4,486932
I try to avoid using $mathbb Z_n$ because it causes so much confusion, but if I do use it then I define it to be equal to $mathbb Z/nmathbb Z$. Denoting $bar1$ by $1$ is just a convenient abuse of notation.
– Derek Holt
Aug 17 at 7:39
@DerekHolt Ya actually both notation are the same in the sense that the two groups stated are isomorphic, and denoting $bar1$ by $1$ is simpler.
– Alan Wang
Aug 17 at 9:26
@DerekHolt: Do you think calling the multiplicative identity in a (unitary) ring $1$ is always an "abuse of notation", or is that only when the ring happens to be a quotient of $mathbb Z$?
– Henning Makholm
Aug 17 at 9:27
@HenningMakholm I think perhaps that abuse of notation was not exactly what I meant. I meant a simplification of notation to be used when there is no danger of confusion. So I would denote the multiplicative identity in a ring $R$ by $1_R$, but say we will usually just write $1$. Unfortunately many students do succeed in getting confused!
– Derek Holt
Aug 17 at 10:27
@AlanWang I thought that even if it was a convention to denote the elements of $mathbbZ/nmathbbZ$ simply by $0, 1, ldots, n-1$, the authors would say so before using the convention. And it was actually the case. They mention it on page 10. Thank you.
– kmiyazaki
Aug 17 at 19:32
add a comment |Â
up vote
4
down vote
accepted
As you can see the author is now discussing the subgroups of $G=Z_n$, not $BbbZ/nBbbZ$, that's why different notation is used.
In the book, the author denotes $Z_n=0,1,dots,n-1$ and $BbbZ/nBbbZ=bar0,bar1,dots,overlinen-1$
answered Aug 17 at 1:47
Alan Wang
4,486932
I try to avoid using $mathbb Z_n$ because it causes so much confusion, but if I do use it then I define it to be equal to $mathbb Z/nmathbb Z$. Denoting $bar1$ by $1$ is just a convenient abuse of notation.
– Derek Holt
Aug 17 at 7:39
@DerekHolt Ya actually both notation are the same in the sense that the two groups stated are isomorphic, and denoting $bar1$ by $1$ is simpler.
– Alan Wang
Aug 17 at 9:26
@DerekHolt: Do you think calling the multiplicative identity in a (unitary) ring $1$ is always an "abuse of notation", or is that only when the ring happens to be a quotient of $mathbb Z$?
– Henning Makholm
Aug 17 at 9:27
@HenningMakholm I think perhaps that abuse of notation was not exactly what I meant. I meant a simplification of notation to be used when there is no danger of confusion. So I would denote the multiplicative identity in a ring $R$ by $1_R$, but say we will usually just write $1$. Unfortunately many students do succeed in getting confused!
– Derek Holt
Aug 17 at 10:27
@AlanWang I thought that even if it was a convention to denote the elements of $mathbbZ/nmathbbZ$ simply by $0, 1, ldots, n-1$, the authors would say so before using the convention. And it was actually the case. They mention it on page 10. Thank you.
– kmiyazaki
Aug 17 at 19:32
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
As you can see the author is now discussing the subgroups of $G=Z_n$, not $BbbZ/nBbbZ$, that's why different notation is used.
In the book, the author denotes $Z_n=0,1,dots,n-1$ and $BbbZ/nBbbZ=bar0,bar1,dots,overlinen-1$
answered Aug 17 at 1:47
Alan Wang
4,486932
As you can see the author is now discussing the subgroups of $G=Z_n$, not $BbbZ/nBbbZ$, that's why different notation is used.
In the book, the author denotes $Z_n=0,1,dots,n-1$ and $BbbZ/nBbbZ=bar0,bar1,dots,overlinen-1$
answered Aug 17 at 1:47
Alan Wang
4,486932
edited Aug 17 at 9:23
answered Aug 17 at 1:47
Alan Wang
4,486932
answered Aug 17 at 1:47
Alan Wang
4,486932
answered Aug 17 at 1:47
Alan Wang
4,486932
4,486932
I try to avoid using $mathbb Z_n$ because it causes so much confusion, but if I do use it then I define it to be equal to $mathbb Z/nmathbb Z$. Denoting $bar1$ by $1$ is just a convenient abuse of notation.
– Derek Holt
Aug 17 at 7:39
@DerekHolt Ya actually both notation are the same in the sense that the two groups stated are isomorphic, and denoting $bar1$ by $1$ is simpler.
– Alan Wang
Aug 17 at 9:26
@DerekHolt: Do you think calling the multiplicative identity in a (unitary) ring $1$ is always an "abuse of notation", or is that only when the ring happens to be a quotient of $mathbb Z$?
– Henning Makholm
Aug 17 at 9:27
@HenningMakholm I think perhaps that abuse of notation was not exactly what I meant. I meant a simplification of notation to be used when there is no danger of confusion. So I would denote the multiplicative identity in a ring $R$ by $1_R$, but say we will usually just write $1$. Unfortunately many students do succeed in getting confused!
– Derek Holt
Aug 17 at 10:27
@AlanWang I thought that even if it was a convention to denote the elements of $mathbbZ/nmathbbZ$ simply by $0, 1, ldots, n-1$, the authors would say so before using the convention. And it was actually the case. They mention it on page 10. Thank you.
– kmiyazaki
Aug 17 at 19:32
add a comment |Â
I try to avoid using $mathbb Z_n$ because it causes so much confusion, but if I do use it then I define it to be equal to $mathbb Z/nmathbb Z$. Denoting $bar1$ by $1$ is just a convenient abuse of notation.
– Derek Holt
Aug 17 at 7:39
@DerekHolt Ya actually both notation are the same in the sense that the two groups stated are isomorphic, and denoting $bar1$ by $1$ is simpler.
– Alan Wang
Aug 17 at 9:26
@DerekHolt: Do you think calling the multiplicative identity in a (unitary) ring $1$ is always an "abuse of notation", or is that only when the ring happens to be a quotient of $mathbb Z$?
– Henning Makholm
Aug 17 at 9:27
@HenningMakholm I think perhaps that abuse of notation was not exactly what I meant. I meant a simplification of notation to be used when there is no danger of confusion. So I would denote the multiplicative identity in a ring $R$ by $1_R$, but say we will usually just write $1$. Unfortunately many students do succeed in getting confused!
– Derek Holt
Aug 17 at 10:27
@AlanWang I thought that even if it was a convention to denote the elements of $mathbbZ/nmathbbZ$ simply by $0, 1, ldots, n-1$, the authors would say so before using the convention. And it was actually the case. They mention it on page 10. Thank you.
– kmiyazaki
Aug 17 at 19:32
I try to avoid using $mathbb Z_n$ because it causes so much confusion, but if I do use it then I define it to be equal to $mathbb Z/nmathbb Z$. Denoting $bar1$ by $1$ is just a convenient abuse of notation.
– Derek Holt
Aug 17 at 7:39
I try to avoid using $mathbb Z_n$ because it causes so much confusion, but if I do use it then I define it to be equal to $mathbb Z/nmathbb Z$. Denoting $bar1$ by $1$ is just a convenient abuse of notation.
– Derek Holt
Aug 17 at 7:39
@DerekHolt Ya actually both notation are the same in the sense that the two groups stated are isomorphic, and denoting $bar1$ by $1$ is simpler.
– Alan Wang
Aug 17 at 9:26
@DerekHolt Ya actually both notation are the same in the sense that the two groups stated are isomorphic, and denoting $bar1$ by $1$ is simpler.
– Alan Wang
Aug 17 at 9:26
@DerekHolt: Do you think calling the multiplicative identity in a (unitary) ring $1$ is always an "abuse of notation", or is that only when the ring happens to be a quotient of $mathbb Z$?
– Henning Makholm
Aug 17 at 9:27
@DerekHolt: Do you think calling the multiplicative identity in a (unitary) ring $1$ is always an "abuse of notation", or is that only when the ring happens to be a quotient of $mathbb Z$?
– Henning Makholm
Aug 17 at 9:27
@HenningMakholm I think perhaps that abuse of notation was not exactly what I meant. I meant a simplification of notation to be used when there is no danger of confusion. So I would denote the multiplicative identity in a ring $R$ by $1_R$, but say we will usually just write $1$. Unfortunately many students do succeed in getting confused!
– Derek Holt
Aug 17 at 10:27
@HenningMakholm I think perhaps that abuse of notation was not exactly what I meant. I meant a simplification of notation to be used when there is no danger of confusion. So I would denote the multiplicative identity in a ring $R$ by $1_R$, but say we will usually just write $1$. Unfortunately many students do succeed in getting confused!
– Derek Holt
Aug 17 at 10:27
@AlanWang I thought that even if it was a convention to denote the elements of $mathbbZ/nmathbbZ$ simply by $0, 1, ldots, n-1$, the authors would say so before using the convention. And it was actually the case. They mention it on page 10. Thank you.
– kmiyazaki
Aug 17 at 19:32
@AlanWang I thought that even if it was a convention to denote the elements of $mathbbZ/nmathbbZ$ simply by $0, 1, ldots, n-1$, the authors would say so before using the convention. And it was actually the case. They mention it on page 10. Thank you.
– kmiyazaki
Aug 17 at 19:32
add a comment |Â
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2
It's very common practice. Also, it is aligned with defining $n:=overbrace1+dots+1^n$ where $1$ is the multiplicative identity.
– Berci
Aug 17 at 1:22
2
Try typsetting a paper or 200-page thesis with bars everywhere. You'll come around.
– Randall
Aug 17 at 1:26