Proving $X/Ycong mathbb{Z}_p^times$
$begingroup$
I would like to show that $X=GA_1(mathbb{Z}_p)$ has a normal subgroup $Y$ that is isomorphic to $mathbb{Z}_p$, that $X/Ycong mathbb{Z}_p^times$ but $Xnotcong mathbb{Z}_ptimes mathbb{Z}_p^times $.
$GA(mathbb{Z}_p)$ is the general affine group of matrices $1times1$ over the field $mathbb{Z}$ mod $p$ (prime) (link). Also $mathbb{Z}_p^times$ is the Multiplicative group of integers modulo $p$ (link).
I know that in order to prove $X/Ycong mathbb{Z}_p^times$ I need to find $Y$ so there is $phi : X/Y to mathbb{Z}_p^times$. But I can't think of $Y$ and $phi$ in order to solve this problem. Which group $Y$ and function $phi$ should solve it and how?
abstract-algebra group-theory
asked Jan 12 at 17:13
vesiivesii
3978
$endgroup$
add a comment |
$begingroup$
I would like to show that $X=GA_1(mathbb{Z}_p)$ has a normal subgroup $Y$ that is isomorphic to $mathbb{Z}_p$, that $X/Ycong mathbb{Z}_p^times$ but $Xnotcong mathbb{Z}_ptimes mathbb{Z}_p^times $.
$GA(mathbb{Z}_p)$ is the general affine group of matrices $1times1$ over the field $mathbb{Z}$ mod $p$ (prime) (link). Also $mathbb{Z}_p^times$ is the Multiplicative group of integers modulo $p$ (link).
I know that in order to prove $X/Ycong mathbb{Z}_p^times$ I need to find $Y$ so there is $phi : X/Y to mathbb{Z}_p^times$. But I can't think of $Y$ and $phi$ in order to solve this problem. Which group $Y$ and function $phi$ should solve it and how?
abstract-algebra group-theory
asked Jan 12 at 17:13
vesiivesii
3978
$endgroup$
add a comment |
$begingroup$
I would like to show that $X=GA_1(mathbb{Z}_p)$ has a normal subgroup $Y$ that is isomorphic to $mathbb{Z}_p$, that $X/Ycong mathbb{Z}_p^times$ but $Xnotcong mathbb{Z}_ptimes mathbb{Z}_p^times $.
$GA(mathbb{Z}_p)$ is the general affine group of matrices $1times1$ over the field $mathbb{Z}$ mod $p$ (prime) (link). Also $mathbb{Z}_p^times$ is the Multiplicative group of integers modulo $p$ (link).
I know that in order to prove $X/Ycong mathbb{Z}_p^times$ I need to find $Y$ so there is $phi : X/Y to mathbb{Z}_p^times$. But I can't think of $Y$ and $phi$ in order to solve this problem. Which group $Y$ and function $phi$ should solve it and how?
abstract-algebra group-theory
asked Jan 12 at 17:13
vesiivesii
3978
$endgroup$
I would like to show that $X=GA_1(mathbb{Z}_p)$ has a normal subgroup $Y$ that is isomorphic to $mathbb{Z}_p$, that $X/Ycong mathbb{Z}_p^times$ but $Xnotcong mathbb{Z}_ptimes mathbb{Z}_p^times $.
$GA(mathbb{Z}_p)$ is the general affine group of matrices $1times1$ over the field $mathbb{Z}$ mod $p$ (prime) (link). Also $mathbb{Z}_p^times$ is the Multiplicative group of integers modulo $p$ (link).
I know that in order to prove $X/Ycong mathbb{Z}_p^times$ I need to find $Y$ so there is $phi : X/Y to mathbb{Z}_p^times$. But I can't think of $Y$ and $phi$ in order to solve this problem. Which group $Y$ and function $phi$ should solve it and how?
abstract-algebra group-theory
abstract-algebra group-theory
asked Jan 12 at 17:13
vesiivesii
3978
asked Jan 12 at 17:13
vesiivesii
3978
asked Jan 12 at 17:13
vesiivesii
3978
asked Jan 12 at 17:13
vesiivesii
3978
asked Jan 12 at 17:13
vesiivesii
3978
3978
add a comment |
add a comment |
1 Answer
1
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oldest
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$begingroup$
$newcommand{Set}[1]{left{ #1 right}}$
It is convenient to think of $X$ as the group of matrices
$$
Set{begin{bmatrix}a & b\
0 & 1end{bmatrix}
a in mathbb{Z}_p^{times}, b in mathbb{Z}_p}.
$$
Hint 1
To find $Y$, take the elements with $a = 1$.
Hint 2
To find $phi$, consider the map that takes the matrix $begin{bmatrix}a & b\0 & 1end{bmatrix}$ to $a$.
answered Jan 12 at 17:32
Andreas CarantiAndreas Caranti
57.3k34497
$endgroup$
$begingroup$
By your hint we get $Y=left{ begin{pmatrix}1 & b\ 0 & 1 end{pmatrix}|binmathbb{Z}_{p}right}$ but why $Y=mathbb{Z}_p$ and $X/Ycongmathbb{Z}_{p}^{times}$?
$endgroup$
– vesii
Jan 12 at 17:43
add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
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active
oldest
votes
$begingroup$
$newcommand{Set}[1]{left{ #1 right}}$
It is convenient to think of $X$ as the group of matrices
$$
Set{begin{bmatrix}a & b\
0 & 1end{bmatrix}
a in mathbb{Z}_p^{times}, b in mathbb{Z}_p}.
$$
Hint 1
To find $Y$, take the elements with $a = 1$.
Hint 2
To find $phi$, consider the map that takes the matrix $begin{bmatrix}a & b\0 & 1end{bmatrix}$ to $a$.
answered Jan 12 at 17:32
Andreas CarantiAndreas Caranti
57.3k34497
$endgroup$
$begingroup$
By your hint we get $Y=left{ begin{pmatrix}1 & b\ 0 & 1 end{pmatrix}|binmathbb{Z}_{p}right}$ but why $Y=mathbb{Z}_p$ and $X/Ycongmathbb{Z}_{p}^{times}$?
$endgroup$
– vesii
Jan 12 at 17:43
add a comment |
$begingroup$
$newcommand{Set}[1]{left{ #1 right}}$
It is convenient to think of $X$ as the group of matrices
$$
Set{begin{bmatrix}a & b\
0 & 1end{bmatrix}
a in mathbb{Z}_p^{times}, b in mathbb{Z}_p}.
$$
Hint 1
To find $Y$, take the elements with $a = 1$.
Hint 2
To find $phi$, consider the map that takes the matrix $begin{bmatrix}a & b\0 & 1end{bmatrix}$ to $a$.
answered Jan 12 at 17:32
Andreas CarantiAndreas Caranti
57.3k34497
$endgroup$
$begingroup$
By your hint we get $Y=left{ begin{pmatrix}1 & b\ 0 & 1 end{pmatrix}|binmathbb{Z}_{p}right}$ but why $Y=mathbb{Z}_p$ and $X/Ycongmathbb{Z}_{p}^{times}$?
$endgroup$
– vesii
Jan 12 at 17:43
add a comment |
$begingroup$
$newcommand{Set}[1]{left{ #1 right}}$
It is convenient to think of $X$ as the group of matrices
$$
Set{begin{bmatrix}a & b\
0 & 1end{bmatrix}
a in mathbb{Z}_p^{times}, b in mathbb{Z}_p}.
$$
Hint 1
To find $Y$, take the elements with $a = 1$.
Hint 2
To find $phi$, consider the map that takes the matrix $begin{bmatrix}a & b\0 & 1end{bmatrix}$ to $a$.
answered Jan 12 at 17:32
Andreas CarantiAndreas Caranti
57.3k34497
$endgroup$
$newcommand{Set}[1]{left{ #1 right}}$
It is convenient to think of $X$ as the group of matrices
$$
Set{begin{bmatrix}a & b\
0 & 1end{bmatrix}
a in mathbb{Z}_p^{times}, b in mathbb{Z}_p}.
$$
Hint 1
To find $Y$, take the elements with $a = 1$.
Hint 2
To find $phi$, consider the map that takes the matrix $begin{bmatrix}a & b\0 & 1end{bmatrix}$ to $a$.
answered Jan 12 at 17:32
Andreas CarantiAndreas Caranti
57.3k34497
edited Jan 12 at 17:34
answered Jan 12 at 17:32
Andreas CarantiAndreas Caranti
57.3k34497
answered Jan 12 at 17:32
Andreas CarantiAndreas Caranti
57.3k34497
answered Jan 12 at 17:32
Andreas CarantiAndreas Caranti
57.3k34497
57.3k34497
$begingroup$
By your hint we get $Y=left{ begin{pmatrix}1 & b\ 0 & 1 end{pmatrix}|binmathbb{Z}_{p}right}$ but why $Y=mathbb{Z}_p$ and $X/Ycongmathbb{Z}_{p}^{times}$?
$endgroup$
– vesii
Jan 12 at 17:43
add a comment |
$begingroup$
By your hint we get $Y=left{ begin{pmatrix}1 & b\ 0 & 1 end{pmatrix}|binmathbb{Z}_{p}right}$ but why $Y=mathbb{Z}_p$ and $X/Ycongmathbb{Z}_{p}^{times}$?
$endgroup$
– vesii
Jan 12 at 17:43
$begingroup$
By your hint we get $Y=left{ begin{pmatrix}1 & b\ 0 & 1 end{pmatrix}|binmathbb{Z}_{p}right}$ but why $Y=mathbb{Z}_p$ and $X/Ycongmathbb{Z}_{p}^{times}$?
$endgroup$
– vesii
Jan 12 at 17:43
$begingroup$
By your hint we get $Y=left{ begin{pmatrix}1 & b\ 0 & 1 end{pmatrix}|binmathbb{Z}_{p}right}$ but why $Y=mathbb{Z}_p$ and $X/Ycongmathbb{Z}_{p}^{times}$?
$endgroup$
– vesii
Jan 12 at 17:43
add a comment |
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