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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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The rank of Sylow subgroup of special linear groups over finite fields

Let $p,\ell$ be two primes, and let $\mathbb{F}_{q}$ be a finite field of order $q=p^r$. We define the rank of a finite group $G$ to the smallest cardinality of a generating set for $G$. We denote by $...
stupid boy's user avatar
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What is the topology of the zero set of this quadratic form?

Consider a quadratic form in $\mathbb{R}^4$ defined as $q(x) = x_1^2 + x_2^2 - x_3^2 - x_4^2$. I am trying to gain as much intuition as possible about the set of vectors such that $q(x) = 0$. One ...
Leo's user avatar
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Condition on finitely generated subgroup of $GL_2(\mathbb{Q})$ to be free

I am considering finitely generated subgroup $G=\langle A,B,C\rangle$ of $GL_2(\mathbb{Q})$ such that $A,B,C$ all have the upper triangular form $$A=\begin{pmatrix}a_1 & a_2 \\ 0 & a_3\end{...
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Are the Euclidean and orthogonal groups locally isomorphic?

I appear to have proven a very counterintuitive result. I would like it if someone could confirm my reasoning. The Euclidean group $\mathrm{Euc}(n)$ is made up of matrices of the form $ \begin{pmatrix}...
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Can we find a normal nilpotent subgroup $G$ of monomial subgroup of $GL_n(D)$ such that $F[G]=M_n(D)$?

Let $D$ be a non-commutative division ring of finite dimension over its center $F$. Also, $n>1$ is a natural number. Consider that $M$ be the monomial subgroup of $GL_n(D)$(containing $n \times n$ ...
Reza Fallah Moghaddam's user avatar
2 votes
1 answer
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A generalization of Baumslag-Solitar groups

I am wondering about the following generalization of the group $B_{1,2}=\langle a,b\, |\, bab^{-1}=a^2\rangle$: $$ G_k=\langle a_1,a_2,\ldots,a_{k+1}\, |\, a_{i+1}a_ia_{i+1}^{-1}=a_i^2, i=1,2,\ldots,k\...
QMath's user avatar
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Why is $O(n)$ not the double cover of $SO(n)$?

$O(n)$ has two connected components, $(\det)^{-1}(1)$ and $(\det)^{-1}(-1)$. While I know it is not true, I am wondering why the above is not enough to say that $O(n)$ is the double cover of $SO(n)$ ...
CBBAM's user avatar
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What does the quotient space $\operatorname{SL}(n) / \sim$ look like? Is it a quotient manifold?

First of all, I have never taken a course about Lie algebra or Riemannian manifold, so please be kind about any of my inappropriate way of naming things or giving bad expression. Suppose $n \in Z^+$, ...
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Alternative description of generalized orthogonal group.

I am studying Lie groups and Lie algebras.One of the standard examples of Lie groups is $O(n;k)$,which is called the generalized orthogonal group.It is defined by $O(n;k)=\{T\in GL_n(\mathbb C): [Tx,...
Kishalay Sarkar's user avatar
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The order of $H = \{X \in \mathrm{GL}_2(\mathbb{Z}_p) \mid \det(X) = 1\}$ with the definition

Let be a prime $p$ and $H = \{X \in \mathrm{GL}_2(\mathbb{Z}_p) \mid \det(X) = 1\}$. I know that the order of a element $g \in \mathrm{GL}_2(\mathbb{Z}_p)$ is the less $k$ such that $g^k = e$, but I ...
Wellington Silva's user avatar
1 vote
1 answer
114 views

Generators for ${\rm SL}_2(\mathbb{Z}/8\mathbb{Z})$ [closed]

Let ${\rm SL}_2(\mathbb{Z}/8\mathbb{Z})$ be the special linear group over the finite ring $\mathbb{Z}/8\mathbb{Z}$. By How to calculate |SL2(Z/NZ)| , we know that the order of the group ${\rm SL}_2(\...
stupid boy's user avatar
2 votes
1 answer
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Prove that every finite subgroup of $SL_2(\mathbb{C})$ is conjugate to a finite subgroup of $SU_2$

I want to prove this proposition. For any finite subgroup $G$ of $SL_2(\mathbb{C})$, I showed that $G$ is unitary with respect to an inner product that we define properly. But I couldn't find the ...
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Injectivity of special orthogonal group

Let $GL_{n}(\mathbb{R})=\{A_{n\times n}\mid A \text{ invertible matrice}\}$, $SL_{n}(\mathbb{R})=\{A\in GL_{n}(\mathbb{R})\mid det(A)=1\}$ be a special linear group and $SO(n)=\{A\in O(n)\mid det(A)=1\...
YSA's user avatar
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Showing that two orbit spaces are isomorphic

Consider the following two group actions: $O_n\times Sym_n(\mathbb{R})\rightarrow Sym_n(\mathbb{R})$ Given by $(A,B)\mapsto ABA^{-1}$, where $O_n$ denotes the group of orthogonal matrices and $Sym_n(\...
Asvr_esn's user avatar
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Proving certain triangle groups are infinite

[Cross-posted to MathOverflow] Consider the Von Dyck group $$ G = \langle x,y\mid x^a=y^b=(xy)^c=1\rangle $$ where $a,b,c\ge3$. Because $G$ is infinite and residually finite, it has an infinite family ...
Steve D's user avatar
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5 votes
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When is general linear group isomorphic to special linear group?

Can $\operatorname{GL}_n(\Bbb F_{q})$ be isomorphic to $\operatorname{SL}_m(\Bbb F_{r})$, where $\Bbb F_q$, $\Bbb F_r$ are finite fields with $q,r$ elements respectively? By considering the center, $$\...
user108580's user avatar
4 votes
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
2 votes
0 answers
64 views

Rational representations and Chevalley bases

Let $G$ be a semi-simple linear algebraic group over $\mathbb{C}$ and $\rho\colon G \to \operatorname{GL}_{\mathbb{C}}(V)$ be a finite-dimesional irreducible rational representation of $G$. Being a ...
Henrique Augusto Souza's user avatar
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the hardness of Conjugacy Search Problem in matrix groups

I learned that the Conjugacy Search Problem is considered as a mathematically hard problem to solve and can be used for cryptography. Conjugacy search problem: Let G be a non-abelian group. Let g,h∈G ...
user1192869's user avatar
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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
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Describe the double coset $H_1QH_1$ in $SO_3$ geometrically.

This is an exercise encountered in the book Algebra by M.Artin. Let $\mathbb{R}^3$ be the three-dimensional Euclidean space with coordinates $(x_1, x_2, x_3). $ Denote $H_i$ the subgroup of $SO_3$ ...
Tsoshamry's user avatar
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Semilinear subgroup $\Gamma L(1,2^n)$ of group $GL(n,2)$

Let $\phi$ be an automorphizm of $\mathbb{F}_{2^n}$ and $a \in \mathbb{F}_{2^n}$. Let's consider a set of functions $\Gamma L(1,2^n) = \{g_{a,\phi} | a\in\mathbb{F}_{2^n}, \phi \in Aut(\mathbb{F}_{2^n}...
Orel_Algebraist's user avatar
2 votes
0 answers
71 views

U(n) is compact and algebraic, but not abelian—why not a contradiction?

the subgroup of unitary matrices $\text{U}(n) \subset GL(n, \mathbb{C})$ is compact and definitely algebraic, with an algebraic group law; on the other hand, it's not abelian. why is this not a ...
BD107's user avatar
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Identifying Particular Matrix Ring

In the course of a big perturbation calculation I was doing, a particular matrix ring kept popping up, whose nice properties were essential to being able to complete the calculation. As such, I've ...
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What is the group generated by all reflections around arbitrary lines in 3D space?

A rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of ...
Reza Fallah Moghaddam's user avatar
-1 votes
1 answer
54 views

why square matrix is group for multipy

There are 4 requirements in order to be group for $$(R^{n*n},*), n\in{N}$$ Closer $ \forall{A,B}\in{R^{n*n}}:A*B\in{R^{n*n}}$ Associativity $\forall{A,B,C}\in{R^{n*n}}:A*(B*C)=(A*B)*C$ Neutral ...
KIM CHANGJUN's user avatar
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1 answer
211 views

Are normal subgroups of a matrix group a matrix group?

I came across this question but I'm not sure how to approach it; My thought process is that by definition all subgroups are groups, then why would it be a different case for matrices? What are some ...
gradual_gradient's user avatar
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Hyperplanes in $SL_2(\mathbb{R})$ containing conjugacy classes of matrices

Let $SL_2(K)$ be the special linear group of rank 2 over a field $K$, as an affine group scheme cut out by the equation $ad - bc = 1$ in $\mathbb{A}^4_K$. Let $H$ be an arbitrary, homogeneous, ...
kindasorta's user avatar
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4 votes
1 answer
156 views

Are $S^1$ and $O_2(\mathbb{R})$ isomorphic?

Let $S^1 = \{z \in \mathbb{C} | |z| = 1\}$ and $O_2(\mathbb{R}) = \{A \in GL_2(\mathbb{R})|A^TA = I_2 \}$. Is the group $(S^1, \cdot)$ isomorphic to $(O_2(\mathbb{R}), \cdot)$? Prove it. This is a ...
Bernardo Maia's user avatar
0 votes
2 answers
401 views

Jordan form of an orthogonal matrix

Let $A$ be an orthogonal matrix. Then there exists $Q$ also orthogonal such that $QAQ^* = D$ for some diagonal matrix $D$. Following this post:Elements of SO(n) is block-diagonalizable I'm not sure ...
Toasted_Brain's user avatar
4 votes
1 answer
291 views

Why can the Orthogonal group be split up in this way?

In my groups course at university, we’ve spent a while on the orthogonal group, leading up the the conclusion that $$ \mathrm{O}_n = \mathrm{SO}_n \mathbin{\dot{\cup}} \begin{pmatrix} -1 ...
Reuben Price's user avatar
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0 answers
189 views

Two non-identity elements in PSL(2,C) commute iff

I know non-identity 2 elements in $PSL(2,\mathbb{R})$ commutes iff they have the same fixed points in $\hat{\mathbb{C}}$. But for $PSL(2,\mathbb{C})=Isom^{+}(\mathbb{H}^3)$, it seems a bit tricky for ...
WaoaoaoTTTT's user avatar
1 vote
1 answer
425 views

Left Invariant Vector field of matrix Lie group

In Hamilton's "Mathematical Gauge Theory" I am trying to understand this proof for the left invariant vector fields of the group $G = GL(d, \mathbb{R})$. I understand how we can assume $X\in ...
Kevin Guo's user avatar
  • 113
-1 votes
1 answer
64 views

Generation a subgroup of $GL_2(\mathbb{C})$

Let $G$ be a group and let $S$ be a nonempty subset of a group $G$. The subgroup of $G$ generated by $S$, which is denoted by $\langle S\rangle$, is equal to the set of all elements of $G$ that can be ...
Marja's user avatar
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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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1 vote
0 answers
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Showing that the identity (path) component of a matrix group is a matrix group

Consider a (not necessarily connected) matrix group $G$, and define the identity component $G^0$ of $G$ as the set of all the elements $g$ of $G$ for which there exists a path $\gamma\colon [0,1]\to G$...
P3p3O's user avatar
  • 155
3 votes
1 answer
162 views

Does a given subset of an infinite group generate a finite or infinite subgroup?

Consider a fixed infinite group, such as $SL(n, \mathbb{R})$. Let $S$ be a given finite subset of the elements of $SL(n, \mathbb{R})$, further suppose that $S$ is closed under inverses. Is there any ...
chaiKaram's user avatar
1 vote
0 answers
304 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 ...
Dick Grayson's user avatar
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0 votes
0 answers
54 views

Show that group of matrices is not a Lie group [duplicate]

This example is from Tapp's book "Matrix Groups for Undergraduates": I'm trying to show that $G$ does not have the structure of a manifold, and thus cannot be a Lie group. Any neighborhood ...
Steppewolf's user avatar
-1 votes
1 answer
38 views

Notation question: $SL(2,3)$, $SL_2(F_3)$, and $SL_2(\mathbb{Z}/3\mathbb{Z})$?

Do this all notation equivalent? $SL(2,3)$, $SL_2(F_3)$, and $SL_2(\mathbb{Z}/3\mathbb{Z})$ I am new to the topic and want to know whether above expressions are same.
RV269's user avatar
  • 29
1 vote
0 answers
88 views

Proving Matrix Lie Groups are Lie Groups without using Lie Algebras

As the title suggests, I'm wondering if there are any proofs that Matrix Lie Groups (defined as closed subgroups of $GL_{n}(\mathbb{C})$) are Lie Groups that do not require the use of Lie algebras. I'...
Thomas Scholey's user avatar
2 votes
1 answer
175 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 ...
zyy's user avatar
  • 977
2 votes
1 answer
139 views

$N^-D N^+$ is Zariski open dense in $\text{GL}(n, \mathbb R)$

Let $N^-$ (and resp. $N^+$) denote the upper (resp. lower) triangular unipotent subgroup of $\text{GL}(n,\mathbb R)$. Let $D$ denote the full diagonal subgroup of $\text{GL}(n,\mathbb R)$ I wonder how ...
No One's user avatar
  • 7,999
1 vote
0 answers
93 views

How to show that $SO_{2,1}^+$ is path connected?

I need to show that $SO_{2,1}^+$ is path connected. First of all here are a couple of definitions we are using in the lecture: Let $n=r+s$ for $r,s \in \mathbb{N}$. $$I_{r,s} := diag(\underbrace{ 1,\...
3nondatur's user avatar
  • 4,200
2 votes
1 answer
307 views

7.5.4 of Stillwell Naive Lie Theory

In section 7.5 of John Stillwell's 'Naive Lie Theory' he constructs a proof of what he calls the 'Tangent space visibility' theorem: that for any path-connected matrix Lie group $G$ with a discrete ...
PlatopusMcJagger's user avatar
1 vote
1 answer
226 views

(Non)compactness of (in)definite unitary groups over $\mathbb{R}$

I have heard and read in various sources that the unitary group $U(p,q)(\mathbb{R})$ is compact if and only if $p = 0$ or $q = 0$. However I don't know of a proof (or a reference for one). Is this ...
stupid_question_bot's user avatar
4 votes
1 answer
248 views

If every finite quotient group of a finitely generated linear group G is solvable, then G is solvable

For this question, I was able to show that each finite quotient is polycyclic: Suppose $N$ is a normal subgroup of finite index. Then, all subgroups of $G/N$ are finite, so $G/N$ is Noetherian. A ...
Algebro1000's user avatar
1 vote
0 answers
153 views

If every finitely generated subgroup of a linear group G is solvable, then G is solvable

I've been working on this question for a long time now and I still have no idea how to approach it. I tried induction on the dimension of the matrices in G but can't complete the induction step. Any ...
Algebro1000's user avatar
0 votes
0 answers
56 views

Is it true that $F^*\wr S_n$ is a solvable group, when $n\leq 5$?

Let $U$ be a linear space over a division ring $D$, $G_1$ a subgroup of $GL(U)$, and $\Gamma$ a subgroup of a symmetric group $S_k$ on $\{1,\ldots,k\}, k>1$. The cartesian product $U^k=V_1$ can ...
Reza Fallah Moghaddam's user avatar
4 votes
2 answers
169 views

Finding the normalizer of $\left\{ \left(\begin{matrix} x &0 \\0 & y \end{matrix}\right) : x,y\in \mathbb R\setminus\{0\} \right\}$

I'm having some trouble with the following question: Let $G=\text{GL}_2(\mathbb R)$. What are the elements of the set: $$N_G \left( \underbrace{\left\{ \left(\begin{matrix} x &0 \\0 & y \end{...
Eduardo Magalhães's user avatar

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