Group Theory (Sylow p-subgroups questions)
Let $GL_nmathbb{F}_p$ be the group of invertible $n times n$ matrices with entries coming from $mathbb{F_p}={0, 1, ..., p-1}$ and with group operation multiplication of matrices. (We are writing $mathbb{F_p}$ for this object to not confuse it with $mathbb{Z_p}$, the $group$ of integers modulo $p$ with the operation $+$ modulo $p$; $mathbb{F_p}$ is the $field$ of order $p$, where $p$ is prime.)
a) Show that $|GL_nmathbb{F_p}|=(p^n-1)(p^n-p)(p^n-p^2)...(p^n-p^{n-1})$
Hint: You need to use the invertible nature of the matrices when its columns are linearly independent.
b) Show that $p cdot p^2 cdot ...cdot p^n$ is the largest power of $p$ dividing the order of $GL_nmathbb{F_p}$
c) Deduce that the matrices $A=(a_{ij})$ with $(a_{ij})=0$ when $i<j$ and $a_{ij}=1$ form a Sylow p-subgroup of $GL_nmathbb{F_p}$.
I gather that we can think of $mathbb{F_p}$ in the same way we think about $mathbb{Q}, mathbb{R}$ and $mathbb{C}$. When we have vectors with entires in $mathbb{F_p}$ we can talk about linear independence of such matrices in the usual way, and matrix multiplication as usual and them being invertible as usual, blah blah blah...
Just getting to grip with Sylow's theorems and this sort of algebra.
group-theory sylow-theory
asked Nov 20 at 18:17
Penguinking14
111
|
show 3 more comments
Let $GL_nmathbb{F}_p$ be the group of invertible $n times n$ matrices with entries coming from $mathbb{F_p}={0, 1, ..., p-1}$ and with group operation multiplication of matrices. (We are writing $mathbb{F_p}$ for this object to not confuse it with $mathbb{Z_p}$, the $group$ of integers modulo $p$ with the operation $+$ modulo $p$; $mathbb{F_p}$ is the $field$ of order $p$, where $p$ is prime.)
a) Show that $|GL_nmathbb{F_p}|=(p^n-1)(p^n-p)(p^n-p^2)...(p^n-p^{n-1})$
Hint: You need to use the invertible nature of the matrices when its columns are linearly independent.
b) Show that $p cdot p^2 cdot ...cdot p^n$ is the largest power of $p$ dividing the order of $GL_nmathbb{F_p}$
c) Deduce that the matrices $A=(a_{ij})$ with $(a_{ij})=0$ when $i<j$ and $a_{ij}=1$ form a Sylow p-subgroup of $GL_nmathbb{F_p}$.
I gather that we can think of $mathbb{F_p}$ in the same way we think about $mathbb{Q}, mathbb{R}$ and $mathbb{C}$. When we have vectors with entires in $mathbb{F_p}$ we can talk about linear independence of such matrices in the usual way, and matrix multiplication as usual and them being invertible as usual, blah blah blah...
Just getting to grip with Sylow's theorems and this sort of algebra.
group-theory sylow-theory
asked Nov 20 at 18:17
Penguinking14
111
That's three questions. Which one do you want to ask?
– Lord Shark the Unknown
Nov 20 at 18:19
Any of them would be great. Should I not have put all three under one question?
– Penguinking14
Nov 20 at 18:23
1
For part (a): The first column can't be the zero vector; that gives you $p^n-1$ possibilities (since there are $p^n$ vectors with $n$ entries). The second column can't be in the linear span of the first column, which has $|mathbb{F}_p|=p$ elements; that means you have $p^n-p$ possibilities for the second column. The third column can't be in the linear span of the first two; that subspace has... etc.
– Arturo Magidin
Nov 20 at 19:51
1
For part (b): use part (a). For part (c), verify that those matrices form a group, and count how many such matrices there are.
– Arturo Magidin
Nov 20 at 19:51
Thanks for the help. I don't see how to use part (a) in part (b). Is it something to do with Lagrange's theorem? And for part (c) can I simply check the group axioms on the matrices to see if it's a group.
– Penguinking14
Nov 21 at 19:42
|
show 3 more comments
Let $GL_nmathbb{F}_p$ be the group of invertible $n times n$ matrices with entries coming from $mathbb{F_p}={0, 1, ..., p-1}$ and with group operation multiplication of matrices. (We are writing $mathbb{F_p}$ for this object to not confuse it with $mathbb{Z_p}$, the $group$ of integers modulo $p$ with the operation $+$ modulo $p$; $mathbb{F_p}$ is the $field$ of order $p$, where $p$ is prime.)
a) Show that $|GL_nmathbb{F_p}|=(p^n-1)(p^n-p)(p^n-p^2)...(p^n-p^{n-1})$
Hint: You need to use the invertible nature of the matrices when its columns are linearly independent.
b) Show that $p cdot p^2 cdot ...cdot p^n$ is the largest power of $p$ dividing the order of $GL_nmathbb{F_p}$
c) Deduce that the matrices $A=(a_{ij})$ with $(a_{ij})=0$ when $i<j$ and $a_{ij}=1$ form a Sylow p-subgroup of $GL_nmathbb{F_p}$.
I gather that we can think of $mathbb{F_p}$ in the same way we think about $mathbb{Q}, mathbb{R}$ and $mathbb{C}$. When we have vectors with entires in $mathbb{F_p}$ we can talk about linear independence of such matrices in the usual way, and matrix multiplication as usual and them being invertible as usual, blah blah blah...
Just getting to grip with Sylow's theorems and this sort of algebra.
group-theory sylow-theory
asked Nov 20 at 18:17
Penguinking14
111
Let $GL_nmathbb{F}_p$ be the group of invertible $n times n$ matrices with entries coming from $mathbb{F_p}={0, 1, ..., p-1}$ and with group operation multiplication of matrices. (We are writing $mathbb{F_p}$ for this object to not confuse it with $mathbb{Z_p}$, the $group$ of integers modulo $p$ with the operation $+$ modulo $p$; $mathbb{F_p}$ is the $field$ of order $p$, where $p$ is prime.)
a) Show that $|GL_nmathbb{F_p}|=(p^n-1)(p^n-p)(p^n-p^2)...(p^n-p^{n-1})$
Hint: You need to use the invertible nature of the matrices when its columns are linearly independent.
b) Show that $p cdot p^2 cdot ...cdot p^n$ is the largest power of $p$ dividing the order of $GL_nmathbb{F_p}$
c) Deduce that the matrices $A=(a_{ij})$ with $(a_{ij})=0$ when $i<j$ and $a_{ij}=1$ form a Sylow p-subgroup of $GL_nmathbb{F_p}$.
I gather that we can think of $mathbb{F_p}$ in the same way we think about $mathbb{Q}, mathbb{R}$ and $mathbb{C}$. When we have vectors with entires in $mathbb{F_p}$ we can talk about linear independence of such matrices in the usual way, and matrix multiplication as usual and them being invertible as usual, blah blah blah...
Just getting to grip with Sylow's theorems and this sort of algebra.
group-theory sylow-theory
group-theory sylow-theory
asked Nov 20 at 18:17
Penguinking14
111
asked Nov 20 at 18:17
Penguinking14
111
asked Nov 20 at 18:17
Penguinking14
111
asked Nov 20 at 18:17
Penguinking14
111
asked Nov 20 at 18:17
Penguinking14
111
111
That's three questions. Which one do you want to ask?
– Lord Shark the Unknown
Nov 20 at 18:19
Any of them would be great. Should I not have put all three under one question?
– Penguinking14
Nov 20 at 18:23
1
For part (a): The first column can't be the zero vector; that gives you $p^n-1$ possibilities (since there are $p^n$ vectors with $n$ entries). The second column can't be in the linear span of the first column, which has $|mathbb{F}_p|=p$ elements; that means you have $p^n-p$ possibilities for the second column. The third column can't be in the linear span of the first two; that subspace has... etc.
– Arturo Magidin
Nov 20 at 19:51
1
For part (b): use part (a). For part (c), verify that those matrices form a group, and count how many such matrices there are.
– Arturo Magidin
Nov 20 at 19:51
Thanks for the help. I don't see how to use part (a) in part (b). Is it something to do with Lagrange's theorem? And for part (c) can I simply check the group axioms on the matrices to see if it's a group.
– Penguinking14
Nov 21 at 19:42
|
show 3 more comments
That's three questions. Which one do you want to ask?
– Lord Shark the Unknown
Nov 20 at 18:19
Any of them would be great. Should I not have put all three under one question?
– Penguinking14
Nov 20 at 18:23
1
For part (a): The first column can't be the zero vector; that gives you $p^n-1$ possibilities (since there are $p^n$ vectors with $n$ entries). The second column can't be in the linear span of the first column, which has $|mathbb{F}_p|=p$ elements; that means you have $p^n-p$ possibilities for the second column. The third column can't be in the linear span of the first two; that subspace has... etc.
– Arturo Magidin
Nov 20 at 19:51
1
For part (b): use part (a). For part (c), verify that those matrices form a group, and count how many such matrices there are.
– Arturo Magidin
Nov 20 at 19:51
Thanks for the help. I don't see how to use part (a) in part (b). Is it something to do with Lagrange's theorem? And for part (c) can I simply check the group axioms on the matrices to see if it's a group.
– Penguinking14
Nov 21 at 19:42
That's three questions. Which one do you want to ask?
– Lord Shark the Unknown
Nov 20 at 18:19
That's three questions. Which one do you want to ask?
– Lord Shark the Unknown
Nov 20 at 18:19
Any of them would be great. Should I not have put all three under one question?
– Penguinking14
Nov 20 at 18:23
Any of them would be great. Should I not have put all three under one question?
– Penguinking14
Nov 20 at 18:23
1
1
For part (a): The first column can't be the zero vector; that gives you $p^n-1$ possibilities (since there are $p^n$ vectors with $n$ entries). The second column can't be in the linear span of the first column, which has $|mathbb{F}_p|=p$ elements; that means you have $p^n-p$ possibilities for the second column. The third column can't be in the linear span of the first two; that subspace has... etc.
– Arturo Magidin
Nov 20 at 19:51
For part (a): The first column can't be the zero vector; that gives you $p^n-1$ possibilities (since there are $p^n$ vectors with $n$ entries). The second column can't be in the linear span of the first column, which has $|mathbb{F}_p|=p$ elements; that means you have $p^n-p$ possibilities for the second column. The third column can't be in the linear span of the first two; that subspace has... etc.
– Arturo Magidin
Nov 20 at 19:51
1
1
For part (b): use part (a). For part (c), verify that those matrices form a group, and count how many such matrices there are.
– Arturo Magidin
Nov 20 at 19:51
For part (b): use part (a). For part (c), verify that those matrices form a group, and count how many such matrices there are.
– Arturo Magidin
Nov 20 at 19:51
Thanks for the help. I don't see how to use part (a) in part (b). Is it something to do with Lagrange's theorem? And for part (c) can I simply check the group axioms on the matrices to see if it's a group.
– Penguinking14
Nov 21 at 19:42
Thanks for the help. I don't see how to use part (a) in part (b). Is it something to do with Lagrange's theorem? And for part (c) can I simply check the group axioms on the matrices to see if it's a group.
– Penguinking14
Nov 21 at 19:42
|
show 3 more comments
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That's three questions. Which one do you want to ask?
– Lord Shark the Unknown
Nov 20 at 18:19
Any of them would be great. Should I not have put all three under one question?
– Penguinking14
Nov 20 at 18:23
1
For part (a): The first column can't be the zero vector; that gives you $p^n-1$ possibilities (since there are $p^n$ vectors with $n$ entries). The second column can't be in the linear span of the first column, which has $|mathbb{F}_p|=p$ elements; that means you have $p^n-p$ possibilities for the second column. The third column can't be in the linear span of the first two; that subspace has... etc.
– Arturo Magidin
Nov 20 at 19:51
1
For part (b): use part (a). For part (c), verify that those matrices form a group, and count how many such matrices there are.
– Arturo Magidin
Nov 20 at 19:51
Thanks for the help. I don't see how to use part (a) in part (b). Is it something to do with Lagrange's theorem? And for part (c) can I simply check the group axioms on the matrices to see if it's a group.
– Penguinking14
Nov 21 at 19:42