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 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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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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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 ...
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(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 ...
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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 ...
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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 ...
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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 ...
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{...