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Questions tagged [linear-groups]

A linear group or matrix group is a group $G$ whose elements are invertible $n \times n$ matrices over a field $F$.

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Embedding the generating set of a finitely generated linear group into $GL_d(\mathbb{Q}[\cdots])$

I am currently studying the text "Systolic growth of Linear Groups" by Bou-Rabbee and Cornulier. In this text they make the claim that, given a finitely generated linear group, you can embed it into $...
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1answer
38 views

Conjugacy classes in space of trace zero 2*2 matrices

I'm trying to find the orbits when $SL_2$ operates by conjugation on $\mathfrak{sl}_2=Lie(SL_2)=\{A|\operatorname{tr} A=0\}$. I have tried to write $X\in sl_2$ and corresponding $AXA^{-1}$ for random ...
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Rebuild a linear algebraic group from its orbits

I have the following situation (following the proof that every linear compact group is algebraic from Vinberg, Gorbatsevich and Onishchik "Lie Groups and Lie algebras III" Chapter 4 Theorem 2.1): Let ...
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41 views

Find group elements of $SL(2,3)$

I am starting to learn about group theory so I am still really new in this field. So, I have the group $SL(2,3)$ with the two generators $$ h = \begin{pmatrix} 0 & a \\ -(1/a) & 0 \end{...
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1answer
73 views

$PSL(2,13)$ has no subgroup of prime index.

I want to show that $PSL(2,13)$ has no subgroup of prime index,where $PSL(2,13) = \frac{SL(2,13)}{\brace-I,I}$. We have the below fact. 《If $G$ be a simple group and $H$ be a subgroup of $G$ such ...
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1answer
38 views

Are Braid groups linear as modules or as Vector spaces?

It is known that Braid groups are linear. I am a bit confused while understanding its linear representation. We say that a group $G$ has linear representation if the map $\alpha: G \rightarrow GL(n,F) ...
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17 views

Given $A \in SL_n(\Bbb R)$ Find $B,P \in SL_n(\Bbb R)$ such that, $A=BPB^{-1}P^{-1}$

I was trying to prove that, $[SL_n(\Bbb R), SL_n(\Bbb R)]=SL_n(\Bbb R)$ My attempt: $[SL_n(\Bbb R), SL_n(\Bbb R)]\subset SL_n(\Bbb R)$ is trivial, trying to prove the other inclusion. So it's ...
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Is $H$ is a normal subgroup of $G$? Yes/NO [closed]

$G = GL_n(\mathbb{R})$ and $H$ is the subgroup of all matrices in $G$ with positive determinant Is $H$ is a normal subgroup of $G$? My attempt : Take G= $ \begin{bmatrix} 2 &0 \\ 0& 1 \...
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1answer
40 views

Proving that $Aw\in \langle w\rangle \implies A$ is of the form $\lambda I_n$

If $w=\begin{bmatrix} w_1\\ \vdots\\w_n \end{bmatrix}$ is a vector in $K^n$ for a field $K$ and $A= \begin{bmatrix} \lambda & & \\ & \ddots & a_{ij}\\ & & \lambda \end{...
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1answer
76 views

Special Linear Group - why positive determinant?

What is the reason for restricting the Special Linear Group $SL_n(\mathbb{C})$ to $\det(A)=+1$? What would be the practical consequence of including $\det(A)=-1$? Someone else answered before with ...
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46 views

Isomorphism $\{1\}\times \mathbb C^*\to \{1\}\times \mathbb C^*$

I am reading a text where I have trouble understanding an argument: Let $f: \mathbb C^*\times \mathbb C^*\to \mathbb C^*\times \mathbb C^*$ an isomorphism, such that $f(\{1\}\times \mathbb C^*)= \{1\}...
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2answers
230 views

Group of matrices form a manifold or euclidean space

There is a very interesting question How can a group of matrices for a manifold. From the answers it looks more like group of matrices form euclidean space than a general manifold. I understand that ...
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1answer
80 views

Proving that a representation is irreducible

Let $V_n = \mathbb{C}^n$ the representations of $SU(n)$ given by matrix multiplication $SU(n) \times V_n \rightarrow V_n, (A, v) \mapsto A \cdot v .\,$ Show that $V_n$ is irreducible. I tried to ...
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1answer
30 views

Algebra of SL(n, K) Lie group

If $M \in SL(n, K)$,then $M$ is a $n\times n$ matrix with entries in $K$ such that $det\ M = +1$. To get the algebra, $\mathfrak sl(n, K)$ I expand as follows (keeping always only first order in $x^a$)...
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123 views

Partitions by a subgroup of $GL_2(\mathbb R)$ of another subgroup of $GL_2(\mathbb R)$

An exercise from Artin's Algebra: Let G and H be the following subgroups of $GL_2(\mathbb R)$: $$G = \{ \begin{bmatrix} x & y\\ 0 & 1 \end{bmatrix} \} , H= \{\begin{bmatrix} x & 0\...
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Prove that $SL(n,\mathbb{Z})$ is generated by $(n^2-n)$ elements.

Statement : Prove that $SL(n,\mathbb{Z})$ is generated by $(n^2-n)$ elements. The determinant is a n linear function of the rows of the matrix. Given any matrix, if the determinant is nonzero, say $...
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90 views

Show that $\text{SO}(2)$ is compact.

Let $\text{Mat}_2(\mathbb{R})$ be the set of $2\times 2$ real matrices with the topology obtained by regarding $\text{Mat}_2(\mathbb{R})$ as $\mathbb{R}^4$. Let $$\text{SO}(2)=\{A\in\text{Mat}_2(\...
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2answers
321 views

Centre of the special linear group $SL_2(\mathbb R)$ or $SL(2,\mathbb R)$

Algebra by Michael Artin Def 2.5.12 Obviously $I$ and $-I$ are in the centre: $AI=IA,A(-I)=(-I)A$. How exactly do I go about doing this? I was thinking to solve for $j,f,g,h$ below $$\begin{...
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40 views

What is the maximal $m$, such that $\mathbb{Z}_2^m \leq GL(n, 2)$?

Is there any closed formula for the function $m(n)$, that is defined as the maximal $m$, such that there is $GL(n, 2)$ has a subgroup isomorphic to $\mathbb{Z}_2^m$? The only things I know currently, ...
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1answer
119 views

Iwasawa Matrix Decomposition Proof

Iwasawa Decomposition (special case): Let $G=SL_n(\Bbb{R})$, $K=$ real unitary matrices, $U=$ upper triangular matrices with $1$'s on the diagonal (called unipotent), and $A=$ diagonal matrices with ...
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References for (especially two-dimensional) general linear groups over *finite* fields

Questions. What are good, citable, detailed sources on general linear groups over finite fields? Especially, $\mathrm{GL}_2(\mathbb{F}_p)$, and most especially, the following: characterization of ...
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66 views

Is it true that $\mathrm{GL}_n(\mathbb{R})=\mathrm{SL}_n(\mathbb{R}) \oplus \{\lambda E : \lambda \in \mathbb{R}^{*}\}$

Does the equality $\mathrm{GL}_n(\mathbb{R})=\mathrm{SL}_n(\mathbb{R}) \oplus \{\lambda E : \lambda \in \mathbb{R}^{*}\}$ hold? I think yes! That's the reason: we have the following exact sequence: $...
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29 views

Abelian group action on a projective space

Let $G$ be an abelian Lie subgroup of the projective linear group $\mathrm{PGL}(n,\mathbb C)=\mathrm {Aut} (\mathbb CP^n) $. Are the orbits of $G$ in $\mathbb CP^n$ closed? (acting by group ...
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36 views

Orbits of a unipotent subgroup

Let $G$ be a connectd linear group which acts transitively on the space $X$. Let $U$ be a unipotent subgroup of $G$. How to prove that $U$-orbits are closed copies of $\mathbb C^k$ in $X$?
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2answers
62 views

Double cover of $SO(2)$

In most textbooks i found that $SPIN(2)$ group is isomorphic to $U(1)$. That is $U(1)$ is double cover of $SO(2)$. But $U(1)$ is isomorphic to $SO(2)$. So how can it be? Does this definition of ...
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1answer
74 views

Isomorphism $SO(3)\cong SU(2)/\{+I,-I\}$: is it just a group isomorphism?

Reading the proof that $SO(3)\cong SU(2)/\{+I,-I\}$, I see that it hinges on the isomorphism theorem for groups. Since I am studying this in a Lie groups course, I would expect these two groups to be ...
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Can the triangularization of a linear group preserve a given good feature of an element?

Let $k$ be an algebraic-closed field and let $G$ be a soluble subgroup of $GL_n(k)$. By a result of Mal'cev (Bertram A.F. Wehrfritz - Infinite Linear Groups, Theorem 3.6), $G$ contains a normal ...
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1answer
93 views

Are the Lie groups $GL_n(\mathbb R)$ and $GL_n(\mathbb C)$ unimodular?

I want to know whether $GL_n(\mathbb R)$ is a unimodular Lie group, that is, whether it has a Haar measure that is both left and right invariant. What I've tried so far is seeing $GL_n(\mathbb R)$ ...
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1answer
46 views

$SU(2)$ acts transitively on $\mathbb{CP}^1$

How can one proof rigorously that the action of $SU(2)$ on $\mathbb{CP}^1$, where $\mathbb{CP}^1$ is the complex projective space, is transitive? i.e., that for any $u, v \in \mathbb{CP}^1$, there ...
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40 views

Help me understand these topological properties of linear groups.

I'm studying Linear groups and already bamboozled after proving that there is a bijective correspondence from $SU_2$to $S^3$ and $SU_2$ can be thought of as the set of unit vectors in the quaternion ...
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1answer
44 views

How to prove that the following subgroup is normal?

Let $M(n,\mathbb{R}$) denote the set of all $n \times n$ matrices with real entries (identified with $\mathbb{R}^{n^2}$ and endowed with its usual topology) and let $GL(n, \mathbb{R})$ denote the ...
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2answers
33 views

Show that the Quotient Group $Γ_2(p)/Γ_2(p^k)$ is finite.

Let $p$ be a prime number. Denote by $Γ_2(p)$ the multiplicative group of all $2×2$ matrices $$ x = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$ with elements $a, b, c, d ∈ \...
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1answer
167 views

A group homomorphism from $GL(n,\Bbb {R})$ such that $SL(n,\Bbb {R})$ is the kernel

I have a confusion about the answer to this question: Every normal subgroup is the kernel of some homomorphism. I was working on the following problem: Define a group homomorphism from $GL(n,\Bbb{...
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3answers
75 views

General affine group of degree one on field of order 4

I want to show $AGL(1,4)$ is isomorphic to $A_4$. I know that $AGL1(k)≅k⋊k^×$, where the semi-direct product is given by the natural action of $k^×$ on $k$ by multiplication. But I dont know how to ...
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3answers
101 views

Prove that $GL_n(\mathbb{C}) $ is not solvable

How should I prove that $GL_n(\mathbb{C}) $, the group of invertible matrices over the complex numbers, is not solvable? I have no idea how to prove this by supposing that $GL_n(\mathbb{C}) $ is ...
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1answer
224 views

Commutator Group of $GL_2(R)$ is $SL_2(R)$

Let $GL_2(R)$ be the general linear group of $2\times2$ matrices and $SL_2(R)$ the special linear group of $2\times2$ matrices. Show that the commutator subgroup of $GL_2(R)$ is $SL_2(R)$. I can show ...
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0answers
15 views

Exactness of the functor $(\quad)^G$ for $G=k^\times$

Suppose $G=k^\times$ where $k$ is an algebraic closed field of characteristic $0$(I'm not sure whether this assumption is necessary). Then show that $(\quad)^G$ is an exact functor i.e. given an exact ...
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2answers
189 views

Why do transformations of ellipses by matrix transformations remain ellipses.

How do I show that given an ellipse which is transformed by a matrix it will remain an ellipse. (unless the matrix is not invert-able). You could consider a line segment or point as a special case of ...
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2answers
202 views

Diagonalizability of elements of finite subgroups of general linear group over an algebraically closed field

How to show that every element of $G$, where $G$ is a finite subgroup of $GL_n(\mathbb{k})$, the general linear group of square matrices of order $n$ over some algebraic closed field $\mathbb{k}$, is ...
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1answer
184 views

General Linear Group $GL_n(\mathbb{Z})$ of Integers is finitely generated

My question refers to following former question of mine: General Linear Group $GL_n(R)$ not Finitely Generated I want to know how to see that the general linear group $GL_n(\mathbb{Z})$ of integers ...
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2answers
116 views

General Linear Group $GL_n(R)$ not Finitely Generated

Let $R$ an infinite field (or more general an infinite ring). Let consider for $n \ge 2$ the general linear group $GL_n(R)$ and the special linear group $SL_n(R)$ considered with canonical matrix ...
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1answer
83 views

The center of a linear group with finite conjugacy classes has finite index

Given a linear group $G\subset GL_n(k)$ with the property that every conjugacy class is finite (or equivalently, each centraliser has finite index), show that it follows that the center has finite ...
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1answer
126 views

Finitely generated torsion subgroup of $SO(3,\mathbb{R})$ is finite

I am aware of a theorem stating that a finitely generated torsion subgroup of $GL_n(\mathbb{C})$ is finite. I am trying to prove a more humble version of the theorem, namely that a finitely generated ...
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93 views

Commutators of matrices and vector fields

The tangent space of $GL(n,\mathbb R)$ in the unit element is $Mat(n,\mathbb R)$ and its elements correspond to left-invariant vector fields on $GL(n,\mathbb R)$. It is well-known that under this ...
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256 views

Centralizer of diagonal matrices in $SL_n(\mathbb{R})$

Let $D$ be the subgroup of $SL_n(\mathbb{R})$ containing all the diagonal matrices (of $SL_n(\mathbb{R})$). I want to find the centralizer of $D$ in $SL_n(\mathbb{R})$, namely $C_{SL_n(\mathbb{R})}(D) ...
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0answers
60 views

Affine quadric surfaces

I want to understand affine quadric surfaces. What are they? and what are their basic properties? Are they homogeneous spaces? And which groups act on them? Thanks for your detailed explanation!
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64 views

Invariants of an action of $\mathsf{SL}_2$ on a space of homogeneous polynomials related to cross-ratio

$\DeclareMathOperator\SL{\mathsf{SL}}$ Let $k$ be a field, let's suppose it's algebraically closed for convenience (take $\operatorname{char}k=0$ if you wish). Consider the (right) group action of $\...
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1answer
152 views

Are finite general linear groups transitive on $F^n$?

Let $F$ be a finite field and let $G=GL_n(F)$ be the general linear group over $F$. There is a group action $G\times F^n \to F^n$ given by $(T,x) \mapsto Tx$. I suspect that this action is not ...
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1answer
129 views

Prove this lie subgroup has finite center

Let $G$ be a connected lie group with finite center. Let $H<G$ be a connected lie subgroup with lie algebra $\mathfrak{h}<\mathfrak{g}$ isomorphic to $\mathfrak{sl}(2,\mathbb{R})$. Prove that $H$...
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0answers
120 views

Maximal compact subgroup of $O(k,m)$

Let $m,n,k$ be nonnegative integers with $m+k=n$. Let $G=\mathrm O(m,k)\subseteq \mathrm{GL}_n(\mathbb{R})$ be the subgroup of matrices preserving the standart non-degenerate symmetric billinear form ...