Calculating a presentation of $mathbb{Z}_{3}$ in detail.
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1
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Theorem: Let $G$ groups and $Ssubset G$ such that
$langle Srangle =G$. (Here $G=left{s_1ldots, s_n:s_iin Scup S^{-1}, ninmathbb{N}right}$.) Let $varphi:Sto G$ with $varphi(s)=s$. By the universal property of free groups there exists a unique homomorphism (in fact, epimorphism) $varphi:F(S)to G$ with $$F(S)=left{win S^{ast}: w text{ reduced word} right}.$$Then $$Gsimeq F(S)/{ker(varphi)}.$$
Here $langle langle Sranglerangle=langle left{gsg^{-1}:sin Scup S^{-1}, gin Gright}rangle.$
Let $Ssubset G$ and $G=langle Srangle.$ Then $langle Smid Trangle $ presentation of $G$ if $G=langle Srangle$ and $Tsubset kervarphi$ and $langle langle Tranglerangle=kervarphi$.
I want prove in a detailed way that $mathbb{Z}_{3}=langle amid a^3rangle.$
I have this.
Here $S=left{aright}.$
First, $a$ must be different from $0$. Because if $a=0$, then $mathbb{Z}_{3}=left{0right}$.
If $a=1,$ then $0=1+1+1, 1=1, 2=1+1$.
If $a=2$, then $0=2+2+2, 1=2+2^{-1}, 2=2$.
Therefore, $mathbb{Z}_{3}=langle arangle$, with $a=1$ or $a=2$.
So . . .
How prove $ker(varphi)=langlelangle left{a^3right}ranglerangle$?
I have this:
$kervarphi=left{s_1cdots s_nin F(S): varphi(s_1cdots s_n)=0, s_iin left{aright}cupleft{a^{-1}right}, ninmathbb{N}right}$
$=left{s_1cdots s_nin F(S): s_1+cdots +s_n=0, s_iin left{aright}cupleft{a^{-1}right}, ninmathbb{N}right}$
group-theory finite-groups cyclic-groups group-presentation universal-property
edited 2 days ago
Shaun
8,205113578
asked Dec 5 at 1:34
eraldcoil
342111
add a comment |
up vote
1
down vote
favorite
Theorem: Let $G$ groups and $Ssubset G$ such that
$langle Srangle =G$. (Here $G=left{s_1ldots, s_n:s_iin Scup S^{-1}, ninmathbb{N}right}$.) Let $varphi:Sto G$ with $varphi(s)=s$. By the universal property of free groups there exists a unique homomorphism (in fact, epimorphism) $varphi:F(S)to G$ with $$F(S)=left{win S^{ast}: w text{ reduced word} right}.$$Then $$Gsimeq F(S)/{ker(varphi)}.$$
Here $langle langle Sranglerangle=langle left{gsg^{-1}:sin Scup S^{-1}, gin Gright}rangle.$
Let $Ssubset G$ and $G=langle Srangle.$ Then $langle Smid Trangle $ presentation of $G$ if $G=langle Srangle$ and $Tsubset kervarphi$ and $langle langle Tranglerangle=kervarphi$.
I want prove in a detailed way that $mathbb{Z}_{3}=langle amid a^3rangle.$
I have this.
Here $S=left{aright}.$
First, $a$ must be different from $0$. Because if $a=0$, then $mathbb{Z}_{3}=left{0right}$.
If $a=1,$ then $0=1+1+1, 1=1, 2=1+1$.
If $a=2$, then $0=2+2+2, 1=2+2^{-1}, 2=2$.
Therefore, $mathbb{Z}_{3}=langle arangle$, with $a=1$ or $a=2$.
So . . .
How prove $ker(varphi)=langlelangle left{a^3right}ranglerangle$?
I have this:
$kervarphi=left{s_1cdots s_nin F(S): varphi(s_1cdots s_n)=0, s_iin left{aright}cupleft{a^{-1}right}, ninmathbb{N}right}$
$=left{s_1cdots s_nin F(S): s_1+cdots +s_n=0, s_iin left{aright}cupleft{a^{-1}right}, ninmathbb{N}right}$
group-theory finite-groups cyclic-groups group-presentation universal-property
edited 2 days ago
Shaun
8,205113578
asked Dec 5 at 1:34
eraldcoil
342111
1
After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark $checkmark$ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?, What should I do if someone answers my question?.
– Shaun
2 days ago
2
There are many different ways of defining $Bbb Z_3$ and a sufficiently detailed answer would depend on which definition you are using. Please be more specific in future.
– Shaun
2 days ago
1
Note, too, that there is no such thing as the presentation of a group, strictly speaking.
– Shaun
2 days ago
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Theorem: Let $G$ groups and $Ssubset G$ such that
$langle Srangle =G$. (Here $G=left{s_1ldots, s_n:s_iin Scup S^{-1}, ninmathbb{N}right}$.) Let $varphi:Sto G$ with $varphi(s)=s$. By the universal property of free groups there exists a unique homomorphism (in fact, epimorphism) $varphi:F(S)to G$ with $$F(S)=left{win S^{ast}: w text{ reduced word} right}.$$Then $$Gsimeq F(S)/{ker(varphi)}.$$
Here $langle langle Sranglerangle=langle left{gsg^{-1}:sin Scup S^{-1}, gin Gright}rangle.$
Let $Ssubset G$ and $G=langle Srangle.$ Then $langle Smid Trangle $ presentation of $G$ if $G=langle Srangle$ and $Tsubset kervarphi$ and $langle langle Tranglerangle=kervarphi$.
I want prove in a detailed way that $mathbb{Z}_{3}=langle amid a^3rangle.$
I have this.
Here $S=left{aright}.$
First, $a$ must be different from $0$. Because if $a=0$, then $mathbb{Z}_{3}=left{0right}$.
If $a=1,$ then $0=1+1+1, 1=1, 2=1+1$.
If $a=2$, then $0=2+2+2, 1=2+2^{-1}, 2=2$.
Therefore, $mathbb{Z}_{3}=langle arangle$, with $a=1$ or $a=2$.
So . . .
How prove $ker(varphi)=langlelangle left{a^3right}ranglerangle$?
I have this:
$kervarphi=left{s_1cdots s_nin F(S): varphi(s_1cdots s_n)=0, s_iin left{aright}cupleft{a^{-1}right}, ninmathbb{N}right}$
$=left{s_1cdots s_nin F(S): s_1+cdots +s_n=0, s_iin left{aright}cupleft{a^{-1}right}, ninmathbb{N}right}$
group-theory finite-groups cyclic-groups group-presentation universal-property
edited 2 days ago
Shaun
8,205113578
asked Dec 5 at 1:34
eraldcoil
342111
Theorem: Let $G$ groups and $Ssubset G$ such that
$langle Srangle =G$. (Here $G=left{s_1ldots, s_n:s_iin Scup S^{-1}, ninmathbb{N}right}$.) Let $varphi:Sto G$ with $varphi(s)=s$. By the universal property of free groups there exists a unique homomorphism (in fact, epimorphism) $varphi:F(S)to G$ with $$F(S)=left{win S^{ast}: w text{ reduced word} right}.$$Then $$Gsimeq F(S)/{ker(varphi)}.$$
Here $langle langle Sranglerangle=langle left{gsg^{-1}:sin Scup S^{-1}, gin Gright}rangle.$
Let $Ssubset G$ and $G=langle Srangle.$ Then $langle Smid Trangle $ presentation of $G$ if $G=langle Srangle$ and $Tsubset kervarphi$ and $langle langle Tranglerangle=kervarphi$.
I want prove in a detailed way that $mathbb{Z}_{3}=langle amid a^3rangle.$
I have this.
Here $S=left{aright}.$
First, $a$ must be different from $0$. Because if $a=0$, then $mathbb{Z}_{3}=left{0right}$.
If $a=1,$ then $0=1+1+1, 1=1, 2=1+1$.
If $a=2$, then $0=2+2+2, 1=2+2^{-1}, 2=2$.
Therefore, $mathbb{Z}_{3}=langle arangle$, with $a=1$ or $a=2$.
So . . .
How prove $ker(varphi)=langlelangle left{a^3right}ranglerangle$?
I have this:
$kervarphi=left{s_1cdots s_nin F(S): varphi(s_1cdots s_n)=0, s_iin left{aright}cupleft{a^{-1}right}, ninmathbb{N}right}$
$=left{s_1cdots s_nin F(S): s_1+cdots +s_n=0, s_iin left{aright}cupleft{a^{-1}right}, ninmathbb{N}right}$
group-theory finite-groups cyclic-groups group-presentation universal-property
group-theory finite-groups cyclic-groups group-presentation universal-property
edited 2 days ago
Shaun
8,205113578
asked Dec 5 at 1:34
eraldcoil
342111
edited 2 days ago
Shaun
8,205113578
asked Dec 5 at 1:34
eraldcoil
342111
edited 2 days ago
Shaun
8,205113578
edited 2 days ago
Shaun
8,205113578
edited 2 days ago
Shaun
8,205113578
8,205113578
asked Dec 5 at 1:34
eraldcoil
342111
asked Dec 5 at 1:34
eraldcoil
342111
asked Dec 5 at 1:34
eraldcoil
342111
342111
1
After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark $checkmark$ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?, What should I do if someone answers my question?.
– Shaun
2 days ago
2
There are many different ways of defining $Bbb Z_3$ and a sufficiently detailed answer would depend on which definition you are using. Please be more specific in future.
– Shaun
2 days ago
1
Note, too, that there is no such thing as the presentation of a group, strictly speaking.
– Shaun
2 days ago
add a comment |
1
After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark $checkmark$ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?, What should I do if someone answers my question?.
– Shaun
2 days ago
2
There are many different ways of defining $Bbb Z_3$ and a sufficiently detailed answer would depend on which definition you are using. Please be more specific in future.
– Shaun
2 days ago
1
Note, too, that there is no such thing as the presentation of a group, strictly speaking.
– Shaun
2 days ago
1
1
After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark $checkmark$ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?, What should I do if someone answers my question?.
– Shaun
2 days ago
After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark $checkmark$ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?, What should I do if someone answers my question?.
– Shaun
2 days ago
2
2
There are many different ways of defining $Bbb Z_3$ and a sufficiently detailed answer would depend on which definition you are using. Please be more specific in future.
– Shaun
2 days ago
There are many different ways of defining $Bbb Z_3$ and a sufficiently detailed answer would depend on which definition you are using. Please be more specific in future.
– Shaun
2 days ago
1
1
Note, too, that there is no such thing as the presentation of a group, strictly speaking.
– Shaun
2 days ago
Note, too, that there is no such thing as the presentation of a group, strictly speaking.
– Shaun
2 days ago
add a comment |
1 Answer
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In order to show $ker(varphi)=langle langle a^3ranglerangle$, one must first be clear on what $varphi$ is, and instead of writing, say, $color{red}{a=1}$, one writes $varphi(a)=1$.
You are on the right track in that you have evidence to suggest that $varphi: amapsto 1text{ or } 2$; that is, that the generator $a$ of the presentation $langle amid a^3rangle$ maps via $varphi$ to one of the elements of $Bbb Z_3$ given by a number coprime to $3$.
Ask yourself,
What reduced words, with letters in ${ a, a^{-1}}$, get mapped to the identity of $Bbb Z_3$ via $varphi$?
But can you deduce what each word maps to under $varphi$ in general?
Hover your cursor over (or, if you're on a touchscreen device, tap) the box below for some hints.
Hint: Use Bézout's Identity, assuming $varphi(a)=p$ for some $p$ coprime to $3$. Here $varphi$ is a homomorphism. Don't forget to show both $ker(varphi)subseteq langlelangle a^3ranglerangle$ and $langlelangle a^3rangleranglesubseteqker(varphi)$.
I hope this helps :)
answered 2 days ago
Shaun
8,205113578
add a comment |
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1 Answer
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In order to show $ker(varphi)=langle langle a^3ranglerangle$, one must first be clear on what $varphi$ is, and instead of writing, say, $color{red}{a=1}$, one writes $varphi(a)=1$.
You are on the right track in that you have evidence to suggest that $varphi: amapsto 1text{ or } 2$; that is, that the generator $a$ of the presentation $langle amid a^3rangle$ maps via $varphi$ to one of the elements of $Bbb Z_3$ given by a number coprime to $3$.
Ask yourself,
What reduced words, with letters in ${ a, a^{-1}}$, get mapped to the identity of $Bbb Z_3$ via $varphi$?
But can you deduce what each word maps to under $varphi$ in general?
Hover your cursor over (or, if you're on a touchscreen device, tap) the box below for some hints.
Hint: Use Bézout's Identity, assuming $varphi(a)=p$ for some $p$ coprime to $3$. Here $varphi$ is a homomorphism. Don't forget to show both $ker(varphi)subseteq langlelangle a^3ranglerangle$ and $langlelangle a^3rangleranglesubseteqker(varphi)$.
I hope this helps :)
answered 2 days ago
Shaun
8,205113578
add a comment |
up vote
2
down vote
In order to show $ker(varphi)=langle langle a^3ranglerangle$, one must first be clear on what $varphi$ is, and instead of writing, say, $color{red}{a=1}$, one writes $varphi(a)=1$.
You are on the right track in that you have evidence to suggest that $varphi: amapsto 1text{ or } 2$; that is, that the generator $a$ of the presentation $langle amid a^3rangle$ maps via $varphi$ to one of the elements of $Bbb Z_3$ given by a number coprime to $3$.
Ask yourself,
What reduced words, with letters in ${ a, a^{-1}}$, get mapped to the identity of $Bbb Z_3$ via $varphi$?
But can you deduce what each word maps to under $varphi$ in general?
Hover your cursor over (or, if you're on a touchscreen device, tap) the box below for some hints.
Hint: Use Bézout's Identity, assuming $varphi(a)=p$ for some $p$ coprime to $3$. Here $varphi$ is a homomorphism. Don't forget to show both $ker(varphi)subseteq langlelangle a^3ranglerangle$ and $langlelangle a^3rangleranglesubseteqker(varphi)$.
I hope this helps :)
answered 2 days ago
Shaun
8,205113578
add a comment |
up vote
2
down vote
up vote
2
down vote
In order to show $ker(varphi)=langle langle a^3ranglerangle$, one must first be clear on what $varphi$ is, and instead of writing, say, $color{red}{a=1}$, one writes $varphi(a)=1$.
You are on the right track in that you have evidence to suggest that $varphi: amapsto 1text{ or } 2$; that is, that the generator $a$ of the presentation $langle amid a^3rangle$ maps via $varphi$ to one of the elements of $Bbb Z_3$ given by a number coprime to $3$.
Ask yourself,
What reduced words, with letters in ${ a, a^{-1}}$, get mapped to the identity of $Bbb Z_3$ via $varphi$?
But can you deduce what each word maps to under $varphi$ in general?
Hover your cursor over (or, if you're on a touchscreen device, tap) the box below for some hints.
Hint: Use Bézout's Identity, assuming $varphi(a)=p$ for some $p$ coprime to $3$. Here $varphi$ is a homomorphism. Don't forget to show both $ker(varphi)subseteq langlelangle a^3ranglerangle$ and $langlelangle a^3rangleranglesubseteqker(varphi)$.
I hope this helps :)
answered 2 days ago
Shaun
8,205113578
In order to show $ker(varphi)=langle langle a^3ranglerangle$, one must first be clear on what $varphi$ is, and instead of writing, say, $color{red}{a=1}$, one writes $varphi(a)=1$.
You are on the right track in that you have evidence to suggest that $varphi: amapsto 1text{ or } 2$; that is, that the generator $a$ of the presentation $langle amid a^3rangle$ maps via $varphi$ to one of the elements of $Bbb Z_3$ given by a number coprime to $3$.
Ask yourself,
What reduced words, with letters in ${ a, a^{-1}}$, get mapped to the identity of $Bbb Z_3$ via $varphi$?
But can you deduce what each word maps to under $varphi$ in general?
Hover your cursor over (or, if you're on a touchscreen device, tap) the box below for some hints.
Hint: Use Bézout's Identity, assuming $varphi(a)=p$ for some $p$ coprime to $3$. Here $varphi$ is a homomorphism. Don't forget to show both $ker(varphi)subseteq langlelangle a^3ranglerangle$ and $langlelangle a^3rangleranglesubseteqker(varphi)$.
I hope this helps :)
answered 2 days ago
Shaun
8,205113578
edited 2 days ago
answered 2 days ago
Shaun
8,205113578
answered 2 days ago
Shaun
8,205113578
answered 2 days ago
Shaun
8,205113578
8,205113578
add a comment |
add a comment |
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After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark $checkmark$ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?, What should I do if someone answers my question?.
– Shaun
2 days ago
2
There are many different ways of defining $Bbb Z_3$ and a sufficiently detailed answer would depend on which definition you are using. Please be more specific in future.
– Shaun
2 days ago
1
Note, too, that there is no such thing as the presentation of a group, strictly speaking.
– Shaun
2 days ago