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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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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