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 $F$.
290 questions
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Is there a non-unimodular group which is not contained in a unimodular group?
The examples I know of non-unimodular (locally compact second countable) groups arise as subgroups the general linear group (e.g. the affine linear group). I was wondering whether there is an example ...
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Why a $\Gamma$ for the general semiaffine group in its notation?
This question is about notation.
I often see the general semiaffine group of degree $n$ over a field $K$ written as $A\Gamma L(n,K)$, $\Gamma A(n,K)$, $A\Gamma L_n(K)$, or $\Gamma A_n(K)$...
Why the ...
- 3,990
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How to Construct a Larger Set of Generators for Sp(2n, q) to Optimize Diameter? [closed]
For the symplectic group Sp(2n, q) over a finite field of order q, I am interested in constructing a set of generators that produces a Cayley graph with a "good diameter." Specifically:
By &...
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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:
...
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The order of a linear profinite group over commutative profinite ring
Recall that one can define the order of a profinite group using the notion of supernatural number. Let $A$ be a commutative profinite ring and let $G$ be a topologically finitely generated closed ...
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Image of Special Orthogonal Groups over a ring $R$ under the map induced by $R \to R/J$.
Let $R$ be a commutative ring with unit and $S \in R^{n \times n}$ a matrix over $R$ such that $S$ is invertible in $R^{n\times n}$ and $S^t = S$. Define $\mathrm{SO}_n(R,S) = \{M \in R^{n \times n} \...
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No continuous isomorphism from $O_4$ to $O_{3,1}$
I am hoping to show that there is no continuous isomorphism from the orthogonal group $O_4$ to the Lorentz group $O_{3,1}$. My approach so far is to make use of the fact that a matrix $A$ is in $O_{3,...
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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 $...
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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 ...
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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
$$A=\begin{pmatrix}a_1 & a_2 \\ 0 & a_3\end{...
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Are the Euclidean and orthogonal groups locally isomorphic?
I appear to have proven a very counterintuitive result. I would like it if someone could confirm my reasoning.
The Euclidean group $\mathrm{Euc}(n)$ is made up of matrices of the form
$
\begin{pmatrix}...
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Can we find a normal nilpotent subgroup $G$ of monomial subgroup of $GL_n(D)$ such that $F[G]=M_n(D)$?
Let $D$ be a non-commutative division ring of finite dimension over its center $F$. Also, $n>1$ is a natural number. Consider that $M$ be the monomial subgroup of $GL_n(D)$(containing $n \times n$ ...
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A generalization of Baumslag-Solitar groups
I am wondering about the following generalization of the group $B_{1,2}=\langle a,b\, |\, bab^{-1}=a^2\rangle$:
$$
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\...
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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 $SO(n)$ ...
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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,...
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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 ...
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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(\...
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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\...
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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(\...
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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 ...
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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, $$\...
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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 ...
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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 ...
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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$....
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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}...
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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 ...
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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 ...
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why square matrix is group for multipy
There are 4 requirements in order to be group
for $$(R^{n*n},*), n\in{N}$$
Closer $ \forall{A,B}\in{R^{n*n}}:A*B\in{R^{n*n}}$
Associativity $\forall{A,B,C}\in{R^{n*n}}:A*(B*C)=(A*B)*C$
Neutral ...
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Are normal subgroups of a matrix group a matrix group?
I came across this question but I'm not sure how to approach it; My thought process is that by definition all subgroups are groups, then why would it be a different case for matrices? What are some ...
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Hyperplanes in $SL_2(\mathbb{R})$ containing conjugacy classes of matrices
Let $SL_2(K)$ be the special linear group of rank 2 over a field $K$, as an affine group scheme cut out by the equation $ad - bc = 1$ in $\mathbb{A}^4_K$.
Let $H$ be an arbitrary, homogeneous, ...
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Are $S^1$ and $O_2(\mathbb{R})$ isomorphic?
Let $S^1 = \{z \in \mathbb{C} | |z| = 1\}$ and $O_2(\mathbb{R}) = \{A \in GL_2(\mathbb{R})|A^TA = I_2 \}$. Is the group $(S^1, \cdot)$ isomorphic to $(O_2(\mathbb{R}), \cdot)$? Prove it.
This is a ...
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Jordan form of an orthogonal matrix
Let $A$ be an orthogonal matrix.
Then there exists $Q$ also orthogonal such that $QAQ^* = D$ for some diagonal matrix $D$.
Following this post:Elements of SO(n) is block-diagonalizable
I'm not sure ...
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Why can the Orthogonal group be split up in this way?
In my groups course at university, we’ve spent a while on the orthogonal group, leading up the the conclusion that
$$
\mathrm{O}_n
=
\mathrm{SO}_n
\mathbin{\dot{\cup}}
\begin{pmatrix}
-1 ...
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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 ...
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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 ...
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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 ...
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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 &...
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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$...
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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 ...
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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 ...
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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 ...
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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.
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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'...
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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 ...
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$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 ...
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0
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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,\...
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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 ...
