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Questions tagged [general-linear-group]

The general linear group of order $n$, $GL_n(\mathbb{F})$, over a field $\mathbb{F}$ (usually $\mathbb{R}$ or $\mathbb{C}$) is the group of $n\times n$ invertible matrices over $\mathbb{F}$. The operation is the usual matrix multiplicatoin.

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The subgroup $B_n \leq \mathrm{GL}_n(F)$ of upper triangular matrices is solvable (Is my proof correct?)

I do my group theory homework again and after some thought I came up with the following proof. It would be nice, if someone could tell me if this is correct. I took some inspiration from this previous ...
Joachim's user avatar
3 votes
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Does every spherical sector of a circle in $\mathrm{M}_n(\mathbb{C})$ contain an invertible matrix?

Some context : (Notations : For the rest of the post, I will identify $\mathrm{M}_n(\mathbb{C})$ with $\mathbb{C}^{n^2}$, and I will note $\mathcal{S}^{n^2-1}$ the unit sphere of $\mathbb{C}^{n^2}$) I ...
Timothe Schmidt's user avatar
1 vote
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References for Maltsev's Theorem on $GL_n(F) \equiv GL_m(K)$ iff $K \equiv F$ and $m=n$.

I have recently found Maltsev's theorem: for $F$ and $K$ algebraically closed fields, $GL_n(F) \equiv GL_m(K)$ if and only if $K \equiv F$ and $m=n$ (thanks to this question: https://mathoverflow.net/...
Natalia Sampedro Loro's user avatar
1 vote
1 answer
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Does there exist an element of order $4$ in $GL_2(\mathbb{Z})/GL_2(\mathbb{Z})'$?

For a group $G$, let $G'$ denote the commutator of the group $G$, and if $H \leq G$ the left cosets will be denoted as $gH$. Now, I understand the fact that $[SL_2(\mathbb{Z}):GL_2(\mathbb{Z})'] = 2.$ ...
Zen's user avatar
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Decomposing (unimodular) matrix over integers into product of matrices mod d

If I have a $n\times n$ unimodular matrix $A \in \text{GL}(n,\mathbb Z)$, i.e., with elements $A_{ij} \in \mathbb Z$, is there some way to decompose the matrix into a product of matrices $A=A^{(1)}A^{(...
Cameron's user avatar
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Parameterization of Irreducible Representations of $\text{GL}_m(\mathbb{C})$

Here is an excerpt from the beginning of Chapter 8 of Fulton's Young Tableaux: . I do not understand how we can parameterize just be diagrams $\lambda$ with at most $m$ rows? What about the length of ...
Anakin Dey's user avatar
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1 answer
91 views

When is the quotient of $\operatorname{GL}_n(\mathbb{R})$ by a discrete subgroup compact?

My question is exactly that on the title. I'm interested in the action of some (discrete) subgroup $H$ on $\operatorname{GL}_n(\mathbb{R})$ by left multiplication. For example, $H$ can be $\...
Bumblebee's user avatar
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Show that as groups, $\mathrm{GL}(V)\cong\mathrm{GL}_n(\mathbb{F})$, with the isomorphism given by $\theta \mapsto A_\theta$

Show that as groups, $\mathrm{GL}(V)\cong\mathrm{GL}_n(\mathbb{F})$, with the isomorphism given by $\theta \mapsto A_\theta$. The notation can be found in my attempt. 1st Attempt: If $\dim_\mathbb{F} ...
Andrés de Fonollosa's user avatar
10 votes
1 answer
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What are the continuous outer automorphisms of the general linear group?

Is the only continuous outer automorphism of $\operatorname{GL}(n, \mathbb{R})$ the transpose inverse map $g \mapsto (g^\intercal)^{-1}$? If not, what other continuous outer automorphisms are there?
Craig's user avatar
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Some property of GL(2,R) and GL(2,Z)

I am trying to show that there exists a family of matrices $(M_n)$ in $GL(2, \mathbb{R}) $ such that $GL(2,\mathbb{Z}) M_n GL(2,\mathbb{Z})$ is the same for every n and $GL(2,\mathbb{Z}) M_n $ is ...
RadonMeasure's user avatar
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1 answer
114 views

Showing that the isomorphism of the general linear group of a vector space with the group of invertible matrices is smooth

This is Example 7.3(e) from John Lee's Introduction to Smooth Manifolds. If $V$ is any real or complex vector space, $GL(V)$ denotes the set of invertible linear maps from $V$ to itself. It is a group ...
nomadicmathematician's user avatar
2 votes
1 answer
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How many elements in $GL_2(\mathbb{F}_p)$ are conjugate to $\begin{pmatrix} \lambda & 0 \\ 0 & \mu\end{pmatrix}$?

How many elements in $GL_2(\mathbb{F}_p)$ are conjugate to $\begin{pmatrix} \lambda & 0 \\ 0 & \mu\end{pmatrix}$ for fixed, distinct $\lambda, \mu \in \mathbb{F}_p$, $p$ prime? I tried arguing ...
Robin's user avatar
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1 vote
1 answer
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Proof that $GL(n,\mathbb{C})$ is isomorphic to a properly embedded Lie subgroup of $GL(2n,\mathbb{R})$

Below is an example of embedded Lie Subgroup from John Lee's Introduction to Smooth Manifolds example 7.18 (d). In this example, why is the image of $\beta$ a properly embedded Lie subgroup of $GL(2n,\...
nomadicmathematician's user avatar
1 vote
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$F \subset E$ as a sub-vector bundle of E, Show that it is a $GL_p$-reduction of $Fr(E)$

Suppose $F \subset E$ is a sub-bundle of E,where one uses "frames of $E$ adapted to $F$", meaning that the first $p$ components of the frame are a frame of $F$ (where we denoted by $r$ and $...
Z.Y.H's user avatar
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1 answer
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Query Regarding the Proof of Z(SO(2m))={±1} in Section 3.7 of 'Naive Lie Theory' by John Stillwell

I have been reading John Stillwell's 'Naive Lie Theory' and in Section 3.7, the author tries to prove that $Z(SO(2m)) = {\pm 1}$. In the proof, a matrix $I^\star$ is introduced with the form: $$ I^\...
Hance Wu's user avatar
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1 answer
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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 ...
user108580's user avatar
1 vote
0 answers
63 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=\...
Dian Wei's user avatar
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2 votes
1 answer
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Fundamental domains for congruence subgroups $\Gamma_0(N)$ by transformations of the fundamental domain of $SL_2(\mathbb{Z})$

I recently started to study conguence subgroups and quotients of the upper half-plane by their action. I found various proofs of the existence of the fundamental region for congruence subgroups that ...
Orazio Cherubini's user avatar
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1 answer
78 views

Perfect Subgroups of $GL(2,p)$, where $p$ is an odd prime. [closed]

It is known that $SL(2,p)$ is a perfect subgroup of $GL(2,p)$ if $p>3$. My question is: Are they the only perfect subgroups of $GL(2,p)$? If not, can $GL(2,p)$ have perfect subgroups whose order is ...
cryptomaniac's user avatar
3 votes
0 answers
79 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 $p$....
Elementary Only's user avatar
8 votes
2 answers
154 views

Does $S_n$ always embed into $GL_{n-1} (\mathbb{F}_p$)?

$S_n$ is the symmetry group of the standard $n-1$-simplex, which is the convex hull of the standard basis vectors in $\mathbb{R}^n$. One can orthogonally project this shape onto the plane $x_1 +...+ ...
GhostNoticer14's user avatar
2 votes
1 answer
90 views

General linear group inclusion

Do we have $\operatorname{GL}(n,F)\le \operatorname{O}(2n,F)$ where O means general orthogonal group and $F$ is an algebraically closed field? I checked some finite group cases: $\operatorname{GL}(2,5)...
scsnm's user avatar
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5 votes
1 answer
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Operatornorm of powers of Matricies with integer coefficient

Let $A\in GL_n(\mathbb{Z})$ have infinite order, so $A^k\neq Id_n$ for all $k>0$. The operator norm is defined by $\lVert A \rVert=\max\{\lVert Av\rVert \mid v\in\mathbb{R}^n: \lVert v\rVert=1\}$. ...
delta's user avatar
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0 answers
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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_{...
nagvalhm's user avatar
1 vote
0 answers
76 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 \\...
nagvalhm's user avatar
2 votes
1 answer
98 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 $R^{*}$ - ...
nagvalhm's user avatar
0 votes
0 answers
47 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 $det$ - ...
nagvalhm's user avatar
0 votes
0 answers
62 views

"Roots" of Unity for the General Linear Group?

I was watching the Michael Penn video about representations when he demonstrated a few matrices that when squared equal the identity (all of them below). These were mostly easy because I could ...
Jenny Pianist's user avatar
2 votes
1 answer
60 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 ...
Zest's user avatar
  • 2,438
0 votes
0 answers
37 views

Representations of general linear Lie algebra vs general linear group [duplicate]

I know that $\mathrm{GL}_{n}(\mathbb{C})$ is not simply connected. Therefore I don’t quite understand the correspondence between representations of $\mathrm{GL}_{n}(\mathbb{C})$ and $\mathfrak{gl}_{n}(...
Matthew Willow's user avatar
1 vote
1 answer
102 views

Why $SL$-invariants and highest weight vectors of rectangular shape coincide?

The groups $\mathrm{SL}(n) := \mathrm{SL}(n,\mathbb{C})$ and $\mathrm{GL}(n) = \mathrm{GL}(n,\mathbb{C})$ acts on $\mathbb{C}^n$ by multiplication from the left. This induces the diagonal action on $\...
SimB4's user avatar
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1 vote
1 answer
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Invertibility of Block Matrix Partial Transpose

Let $$M = \left[\begin{matrix} M_{1,1} & M_{1,2} & \cdots & M_{1,n}\\ M_{2,1} & M_{2,2} & \cdots & M_{2,n}\\ \vdots & \vdots & \ddots & \vdots\\ M_{n,1} & M_{n,...
lily44flying40dw's user avatar
4 votes
1 answer
84 views

Index of an explicit subgroup of $\mathrm{GL}_4(\mathbb{Z})$

Let $H$ be the subgroup of $\mathrm{GL}_4(\mathbb{Z})$ generated by the $4!$ permutation matrices together with $$ \begin{pmatrix} 1 & 0 & 0 & 0 \\ -1 & 0 & 1 & 1 \\ 0 & 1 &...
Joshua P. Swanson's user avatar
4 votes
1 answer
113 views

Principal series representation isomorphism

The problem: Let $G = \mathrm{GL}_2(\mathbb Q_p)$ and $k$ be an algebraically closed field of characteristic $p.$ Denote by $\overline B$ the subgroup of all lower triangular matrices in $G$ and by $...
ayan's user avatar
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0 votes
1 answer
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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) \...
dahemar's user avatar
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3 votes
1 answer
140 views

Is $GL(n)\otimes GL(n)$ is closed in $GL(n^2)$?

Let $GL(n) := \mathrm{GL}(n, \mathbb{C})$ - be space of invertible $n\times n$ matrices over $\mathbb{C}$, i.e. matrix Lie group. Let $H := GL(n) \otimes GL(n)$ be a group with multiplication given by ...
SimB4's user avatar
  • 125
1 vote
0 answers
28 views

Equation in $GL(2,\mathbb{Z}_{2^{n-1}})$

I'm now studying how to embed generalized quaternion group $Q_{2^n}=a^{2^{n-1}}=e$, $u^2=a^{2^{n-2}}$, $ua=a^{-1}u$ in $GL(k,\mathbb{Z}_{2^{n-1}}).$ I got the embedding in case $k=3$, but $k=2$ is ...
Marja's user avatar
  • 37
2 votes
1 answer
189 views

$GL_n(C)$ is isomorphic to a lie subgroup of $GL(2n,R)$

$GL_n(C)$ is isomorphic to a lie subgroup of $GL_{2n}(R)$. I see some posts concerning this (not the same claim): $GL(n, \mathbb{C})$ is isomorphic to a subgroup of $GL(2n, \mathbb{R})$ $GL_n\mathbb{C}...
Mat999's user avatar
  • 537
2 votes
1 answer
65 views

A non-trivial homomorphism from $SL(2, 7)$ to $GL(4, 11)$.

Let $G=GL(4, 11)$ be the general linear group of $4\times 4$ matrices over the field $\mathbb{F}_{11}$. Let $H=SL(2, 7)$ be a special linear group of $2\times 2$ matrices over the field $\mathbb{F}_{...
PAMG's user avatar
  • 4,460
0 votes
1 answer
58 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. ...
Raman's user avatar
  • 189
2 votes
1 answer
110 views

No automorphisms of order $p^2$

Let $H$ be the group of integers mod p, under addition, where $p$ is a prime number. Suppose that n is an integer satisfying $1 ≤ n ≤ p$, and let G be the group $H × H × · · · ×H $($n$ factors). Show ...
nkh99's user avatar
  • 471
0 votes
0 answers
312 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 ...
pyridoxal_trigeminus's user avatar
0 votes
1 answer
146 views

Polar decomposition on $\mathbf{GL}(n,\mathbb{R})$

I am asked to prove that for $g\in\mathbf{GL}(n,\mathbb{R})$, $g$ can be unique written as $$g=k_1 \begin{pmatrix} a_1 & & 0\\ & \ddots &\\ 0 & & a_n \end{...
End points's user avatar
0 votes
1 answer
203 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 ...
Clyde Kertzer's user avatar
0 votes
0 answers
98 views

Chapter 1, Exercise 12 in Brian Hall's Lie groups, Lie algebras, and representations

Suppose $A$ and $B$ are invertible $n \times n$ matrices. Show that there are only finitely many complex numbers $\lambda$ for which $\text{det}( \lambda A + (1-\lambda)B) = 0$. Show that there exists ...
AleNekro97's user avatar
3 votes
0 answers
75 views

Embedding the generalised quaternion group into a general linear group

It's known that there are four non-abelian groups with cyclic subgroup of index $2$. Those groups are the dihedral group $D_{2^n}$, generalised quaternion group $Q_{2^n}$, modular-maximal group $M_{2^...
Marja's user avatar
  • 37
0 votes
0 answers
114 views

Action of $GL(2,\mathbb{Z})$ on lower half plane of $\mathbb{C}$?

I began reading about modular forms and I had a question. So, I know that $SL(2,\mathbb{Z})$ is mapped to the upper half complex plane using a function. The way I see it, the reason we map it to the ...
Perfectoid's user avatar
2 votes
1 answer
138 views

$S_4$ as a subgroup of $GL_3(\mathbb{F}_2)$

I am trying to find a subgroup of $GL_3(\mathbb{F}_2)$, which is isomorphic to $S_4$. Our teacher gave us a hint: we should look at matrices with first column $(1, 0, 0)^T$. But what is the next step? ...
Motoko's user avatar
  • 103
1 vote
1 answer
62 views

$GL(n, \mathbb{C})$ is not residually finite

I saw in some text that $GL(n,\mathbb{C})$ is not residually finite since it is connected. I am not sure about this implication. I guess the proof works somehow like this: we endow $GL(n, \mathbb{C})$ ...
Ja_1941's user avatar
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1 vote
0 answers
58 views

Isomorphism between quternian and SU(2) and their homomorphisms to SO(3)

From Kostrikin, A. I. (1982). Introduction to Algebra. Springer-Verlag, $$\Gamma: \operatorname{SP}(1)\subset\mathbb{H} \to \operatorname{SU}(2)$$ $$a+bi+cj+dk \mapsto \left(\begin{matrix} a+bi &...
Hance Wu's user avatar
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