All Questions
Tagged with general-linear-group abstract-algebra
57 questions
0
votes
0
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47
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Is the Valentiner Group isomorphic to PGL(3, 4)?
In this answer to a question from a while back, it says that the Valentiner group is isomorphic to $PGL_3(\mathbb{F}_4)$. However, when I implement in sage:
...
7
votes
1
answer
104
views
Every finite subgroup of $GL_n(Q)$ is conjugated to a subgroup of $GL_n(R)$.
Claim: let $R$ be a principal ideal domain and $Q$ be its quotient field. Then every finite subgroup of $GL_n(Q)$ is conjugated to a subgroup of $GL_n(R)$.
So I want to solve this exercise but I ...
- 335
3
votes
1
answer
144
views
Is each representation induced by a simpler representation
Let $G$ be a finite group, $V$ be a real vector space of dimension $N$, and $N > |G|$.
Can you find a representation $\sigma: G \to GL(\text{End}(V))$ such that $\sigma$ is not induced by a ...
- 2,596
-1
votes
1
answer
60
views
Quotient group structure of general linear group
I want to find all quotient group structure in $GL(\mathbb{R}^n)$, which is all n*n invertible matrix on real number. The quotient group structure in $GL(\mathbb{R}^n)$ means to find normal subgroup $...
- 94
4
votes
1
answer
129
views
Prove that $\operatorname{GL}_n(k)$ and $\operatorname{SL}_n(k)$ cannot be symmetric group?
Problem. Prove that $\operatorname{GL}_n(k)$ and $\operatorname{SL}_n(k)$ cannot be isomorphic to $S_m$, $m\geq 4$ if $k$ is a finite field with at least two elements.
I am trying to argue by looking ...
- 643
1
vote
0
answers
69
views
is there any unit upper triangular matrix in $GL_n(F_p)$ such that the following holds
Let $U_1$ be the group of $n\times n$ upper triangular matrices with 1's down the main diagonal (called unit upper triangular matrices) over $\mathbb{F}_p$ , which is also a Sylow $p$-subgroup of $G=\...
- 567
3
votes
0
answers
86
views
Number of homomorphisms from $\Bbb Z_p$ to $\operatorname{GL}_2(\Bbb Z_q)$?
I am trying to count the total possible number of group homomorphisms from $\Bbb Z_p$ to $\operatorname{GL}_2(\Bbb Z_q)$, where $p<q$ are primes and $\Bbb Z_p$ denotes the additive group modulo $p$....
2
votes
0
answers
76
views
Emil Artin determinant is unique
In book of Emil Artin "Geometric Algebra" i found such definition of determinant Det: function on matrix rows $A_{1}, ..., A_{n}$, that satisfies:
$Det(A_{1}, ..., b*Ai,..., A_{n})=b*Det(A_{...
- 97
1
vote
0
answers
100
views
Dieudonne determinant is well defined.
Consider $GL_{n}(K)$ over field. By Dieudonne, $\forall A \in GL_{n}(K) \ (A = B*D(x))$ where $B$ is multiplication of some transvections and $D(x)=$
\begin{bmatrix}
1 & 0 &0& \dots \\...
- 97
2
votes
1
answer
104
views
Is determinant canonical projection $\det:GL(n,\mathbb R) \rightarrow GL(n,\mathbb R)^{ab}$?
Is it possible to define determinant as canonical projection from general linear group to its abelianization? Using determinant we can show, that abelianization of $GL(n,R)$ is isomorphic to $R^{*}$ - ...
- 97
0
votes
0
answers
48
views
Any homomorphism from $GL(n, F)$ to $F$ is composition of $det$ and $F$-endomorphism. [duplicate]
I found statement of theorem, that for any field $F$, any homomorphism $f:GL(n, F)\rightarrow F^{*}$ is composition $f=g\circ det$ for some $g:F^{*}\rightarrow F^{*}$ - endomorphism, and $det$ - ...
- 97
2
votes
1
answer
67
views
Linearity of Groups - does it matter which linear groups we consider?
In J. Meier's book "groups, graphs and trees" after remark 3.8 it is stated that
A group that can be faithfully represented as a matrix group is called
a linear group.
Other sources (most ...
- 2,486
0
votes
1
answer
105
views
Let $R$ be a principal ideal domain with field of fractions $K$, then $N_{GL_n(K)}(GL_n(R)) = K^\times GL_n(R)$
Let $R$ be a principal ideal domain with field of fractions $K$. Let $\mathcal{G}_n(K)$ denote the set of subgroups of $GL_n(K)$, where $GL_n(K)$ acts by conjugation on $\mathcal{G}_n(K)$:
$$GL_n(K) \...
- 1,808
0
votes
1
answer
64
views
Generating set of General linear group [closed]
What is one possible minimal generating set of the general linear group $GL_{m}(Z_{p})$?
It might be very easy question whose solution is known to everyone except me. Kindly help me with the same.
...
- 199
1
vote
0
answers
498
views
Prove that Aut($\mathbb Z \times \mathbb Z$) $\cong$ $\text{GL}_2(\mathbb Z)$
Prove that Aut($\mathbb Z \times \mathbb Z$) $\cong$ $\text{GL}_2(\mathbb Z)$.
This is a HW problem for an Algebra course, hints/suggestions welcome.
I didn't find this problem on math.SE, however I ...
- 2,827
0
votes
1
answer
322
views
Proof techniques to show a representation is faithful
I am curious what proof method is most commonly used to show that a representation is faithful. I have found remarkably little online about this question..
It makes sense how to show that a ...
- 3,283
1
vote
0
answers
403
views
Prove that the quaternion group of order 8 is isomorphic to a subgroup of $SL_2(\mathbb{F_3})$ generated by the following two elements.
The Problem: Prove that the subgroup of $SL_2(\mathbb{F_3})$ generated by $\begin{pmatrix}0&-1\\1&0\\\end{pmatrix}$ and $\begin{pmatrix}1&1\\1&-1\\\end{pmatrix}$ is isomorphic to the ...
- 1,487
2
votes
1
answer
208
views
The general linear group $GL(n, \mathbb{C})$ has no proper subgroup of finite index.
Problem: Show that the general linear group $G = GL(n, \mathbb{C})$ has no proper subgroup of finite index.
I wrote down a proof, but not quite sure if it is right, especially about the part about ...
- 1,103
2
votes
1
answer
287
views
Find all subgroups of $\mathrm{GL}(2,\mathbf{R})$ of index 2.
Question: We want to find all subgroups of $\mathrm{GL}(2,\mathbf{R})$ of index $2$.
Here is my first attempt: We already know that for a group $G$, if a subgroup $H$ satisfies $(G\colon H)=2$, then $...
- 123
1
vote
0
answers
88
views
$GL_n(\mathbb{Z}_{pq})$ isomorphic to $GL_n(\mathbb{Z}_{p})\times GL_n(\mathbb{Z}_{q})$
I was trying to solve the problem of finding the number of elements of $GL_n(\mathbb{Z}_m)$ for some $m\in\mathbb{N}$. The case where $m=p$ prime was easy because of elementary linear algebra. The ...
- 2,056
2
votes
2
answers
113
views
Find infinitely many homomorphism from $GL_2(\mathbb Q)$ to $\mathbb Q^*$
Find infinitely many homomorphism from $GL_2(\mathbb Q)$ to $\mathbb Q^*$. Here $\mathbb Q^*$ means the multiplicative group of nonzero rational numbers.
My attempt: An example is the determinant ...
- 802
5
votes
1
answer
125
views
How many conjugacy classes of elements of order 7 are there in $GL(6, \mathbb{F}_2)$?
How many conjugacy classes of elements of order 7 are there in $GL(6, \mathbb{F}_2)$?
I've been thinking the following, every element of order 7 must satisfy the next equation:
$$x^7-1=0$$
We can ...
- 129
0
votes
1
answer
259
views
If every proper subgroup of $G$ is cyclic and normal then is $G$ finite cyclic?
Ok, this may look familiar. For example, here you find a very close question, but not using the normal part of it. This is a exercise from a Brazillian book, Paulo A. Martin's "Grupos, Corpos e ...
0
votes
0
answers
174
views
Automorphisms vs General Linear Group typo
So given the following math question:
Given commutative ring $R$ and $n \in\mathbb{Z^+}$, show that the automorphism group $\text{Aut}(R^n)\cong \text{GL}_n(R)$ (the general linear group).
First, I ...
- 409
-1
votes
1
answer
87
views
Confusion on the order of $GL_2(\mathbb{F}_p)$
In this question, they show that the order of $GL_2(\mathbb{F}_p)$ is $(p^2-1)(p^2-p)$.
For the first column, there are $p^2$ options, and we need to exclude the $0$ column, so there are $p^2-1$. That'...
- 700
1
vote
1
answer
779
views
Unipotent vs nilpotent subgroup of GL(n)
I see the subgroup
$$\left\{\begin{pmatrix}1 & x \\ 0 & 1\end{pmatrix}\right\} \subset \mathrm{GL}(2)$$
written as $U$ and described as the unipotent subgroup of $\mathrm{GL}(2)$ in some ...
- 11
0
votes
0
answers
102
views
Rank of the group of Upper Unitriangular Matrices
Let $F$ be a field of order $p^a$ where $p$ is prime and $a\in \mathbb{N}$. For $n\geq2$ let $P\leq GL_n(F)$ be the group of upper unitriangular matrices, i.e. upper triangular matrices with all ...
- 145
1
vote
0
answers
161
views
Is every element of $G$ of order $p$ conjugate to the following matrix?
For any prime $p,$ consider the group $G = \text {GL}_2 \left (\Bbb Z / p \Bbb Z \right ).$ Then show that every element of $G$ of order $p$ is conjugate to a matrix $\begin{pmatrix} 1 & a \\ 0 &...
- 2,932
-1
votes
1
answer
305
views
How to determine the order of an element of Special Linear Group
Let,
$S=\begin{pmatrix} 0 &-1 \\ 1 & 0 \end{pmatrix}$
and
$T=\begin{pmatrix} 1 &1 \\ 0 & 1 \end{pmatrix}$ .
The matrices $S$ and $T$ generate $SL_2(\mathbb Z)$. The matrix $...
0
votes
1
answer
148
views
Isomorphism between $GL_3(\mathbb{Z}_2)$ and a subgroup of $S_7$
From this question, it is clear that the group of $3\times3$ matrices over $\mathbb{Z}_2$ acts transitively as well as faithfully over the set of $3$-tuples. But how can we conclude that the group is ...
- 7,165
2
votes
1
answer
119
views
Image of a subgroup under the projection to the projective general linear group is isomorphic to the group quotiented by its centre
Let $(V,\rho)$ be a finite, irreducible representation of a finite group $G$, where $V$ is a finite-dimensional vector space. So:
$$
\rho: G \to \textrm{GL}(V)
$$
is a homomorphism.
Let $Z(V)$ denote ...
- 2,907
3
votes
2
answers
127
views
Checking that the given function is really a homomorphism.
Here is the question and its answer:
(a) To what familiar group is $GL_{2}(\mathbb{Z}_2)$ isomorphic?
Answer.
$(a)$
$GL_2(\mathbb{Z}_2)$ is the set of invertible (non-zero determinant) $2\times 2$ ...
user778657
1
vote
1
answer
827
views
conjugacy classes of $GL(3,\mathbb{Z}_p)$
How many conjugacy classes does the group $GL(3,\mathbb{Z}_p)$ have, if $p$ is a prime?
Until now I have a (very broad) lower boud, using that the number of conjugacy classes in a group is at least as ...
- 341
2
votes
1
answer
281
views
Show that the $GL(n,\mathbb R)/P_k$ is isomorphic to the $GL(n,\mathbb R)$-set grassmannian.
Let $r<n$ be two positive integers and $G=GL(n,\mathbb{R}).$ If $Gr(k,\mathbb{R}^n)$ is the set of all $k$-subspaces, then show that the $G$-sets $Gr(k,\mathbb{R}^n)$ and $G/P_k$ is isomorphic, ...
- 169
1
vote
1
answer
519
views
Compute the order of an element in $GL(n, 2)$
Is there an efficient method to compute the order of a matrix $M$ of size $n \times n$ with elements from $GF(2)$ for large (=32,64,128) $n$? I.e. compute the smallest $i$ such that $M^i = I$.
I've ...
- 111
2
votes
1
answer
112
views
$G/F(G)$ is isomorphic to $X_1\times\cdots\times X_t$
Since I still don’t know the answer, I’ve also asked it on math.overflow.
I saw a remark in an old post that
$G/F(G)$ is isomorphic to a group of the form $X_1 \times \ldots \times X_t,$ where ...
user784417
0
votes
2
answers
3k
views
$\{A \in GL_2(\mathbb R):|\det(A)|=1\}$ is a normal subgroup of $GL_2(\mathbb R)$
I have to prove that $O_2(\mathbb R)=\{A \in GL_2(\mathbb R):|\det(A)|=1\}$ is a normal subgroup of $GL_2(\mathbb R)$.
I tried to go on with the definition of normal subgroup but I don't really know ...
- 973
0
votes
1
answer
175
views
Literature for the longest Element of the Weylgroup for $GL(n,K)$. [closed]
I am looking for literature where I can find how the longest Element of the Weyl group looks like for $G=GL(n,K)$ over the diagonal matrices in $G$. I don't even need a proof. But I have no idea where ...
- 13
0
votes
0
answers
106
views
Preimage of soluble subgroup of projective general linear group under canonical projection is soluble
Let $V$ be a vector space and $\textrm{GL}(V)$ the group of automorphisms on $V$.
Let $Z(V) \vartriangleleft \textrm{GL}(V)$ the normal subgroup of homotheties on $V$ (i.e. the centre of $\textrm{GL}(...
- 2,907
0
votes
1
answer
117
views
Cyclic irreducible subgroups of $GL(2,p)$
Let $p$ be an odd prime. Is it possible to have an cyclic irreducible subgroup of $GL(2,p)$ of order $q^n$, say, with $n>0$ and some prime $q$, such that some proper subgroup is still irreducible?
- 4,220
1
vote
1
answer
1k
views
Particular generators of GL(2,R)
I want to prove that $$ \text{GL}_{2}( \mathbb{R} ) = \left\langle \begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 0 \\ 0 & a \end{pmatrix}, \begin{pmatrix} 1 & 0 ...
user761707
4
votes
1
answer
345
views
semidirect product between subgroup of general linear group and vector space in GAP
I am currently working on trying to get a solvable doubly transitive permutation group using GAP. So, I am trying to create the semidirect product of a subgroup of a general linear group and a vector ...
- 43
1
vote
0
answers
65
views
Dimension of orbit of a one-dimensional vector under action of $SO(n)$
Intuitively, I know that if $e_1$ is a standard basis vector in $\mathbb{R}^n$, than multiplying $e_1$ by elements of $SO(n)$ rotates $e_1$ around the $(n-1)$-dimensional sphere.
How do I prove, or ...
- 1,916
1
vote
3
answers
2k
views
$GL(n, \mathbb{C})$ is isomorphic to a subgroup of $GL(2n, \mathbb{R})$
Prove that $GL(n, \mathbb{C})$ is isomorphic to a subgroup of $GL(2n, \mathbb{R})$.
My Proof:
For an $A \in GL(2, \mathbb{C})$, $$ A = \begin{bmatrix} a+bi &c+di \\ e+fi & g+hi \end{bmatrix} ...
- 1,916
2
votes
1
answer
77
views
If $G = GL(2,3)$, $G/Z(G) \cong S_4$
Let $G = GL(2,3)$, the group of all invertible $2 \times 2$ matrices over the field of $3$ elements. Show that $G/Z(G) \cong S_4$.
I know that $G/Z(G)\cong Inn(G)$, where $Inn(G)$ is the inner ...
- 1,523
0
votes
1
answer
198
views
Typo in Algebra by Artin regarding center of the special linear group $SL_n(\mathbb R)$
I saw this question that was asked a year ago: Centre of the special linear group $SL_2(\mathbb R)$ or $SL(2,\mathbb R)$
I will link the photo in question here:
Shouldn't the last line read, "$SL_n(\...
- 1,214
1
vote
1
answer
372
views
How to find a 2-Sylow subgroup of $GL_2(F_3)$?
This question is similar to the question "How to find a p-Sylow subgroup of $GL_2(F_p)$?", which is relatively easy. Since if they share the same prime p, then we can quickly conclude the order of any ...
- 63
4
votes
2
answers
1k
views
The center of the group of $n\times n$ upper triangular matrices with a diagonal of ones
Let $\mathbb{F}_{p}$ be a finite field of order $p$ and $H_{n}(\mathbb{F}_{p})$ be the subgroup of
$GL_n(\mathbb{F}_{p})$ of upper triangular matrices with a diagonal
of ones. Note that the center $Z(...
- 338
0
votes
1
answer
199
views
Cardinality of group generated by two matrices [closed]
What is the cardinality of the group generated by $$\begin{pmatrix}0&-1\\1&0\end{pmatrix},\begin{pmatrix}0&1\\-1&-1\end{pmatrix}$$
under multiplication?
- 71
1
vote
0
answers
184
views
Conjugacy classes of $GL(4,\mathbb{C})$
I have the following question from a past Algebra qualifying exam:
Let $G=GL(4,\mathbb{C})$ be the group of $4\times4$ invertible matrices with complex entries. List in a precise way the conjugacy ...
- 1,035