Mathematics Stack Exchange
Mathematics Stack Exchange

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 $F$.

A matrix group is a group $G$ consisting of invertible matrices over a specified field $F$, with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group.

Any finite group is linear, because it can be realized by permutation matrices using Cayley's theorem.

As examples of linear groups, we have the general linear group, of all invertible $n\times n$ matrices, the special linear group, of all $n\times n$ matrices whose determinant is $1$, or the group of all invertible $n\times n$ upper triangular matrices.

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The Center of $\operatorname{GL}(n,k)$

The given question: Let $k$ be a field and $n \in \mathbb{N}$. Show that the centre of $\operatorname{GL}(n, k)$ is $\lbrace\lambda I\mid λ ∈ k^∗\rbrace$. I have spent a while trying to prove this and have succeeded if $ k \subseteq \mathbb{R}$. So I…
Ben
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Does SO(3) preserve the cross product?

Let $g\in SO(3)$ and $p,q\in\mathbb{R}^3$. I wondered whether it is true that $$g(p\times q)=gp\times gq$$ I am not sure how to prove this. I guess I will use at some point that the last row $g_3$ of $g$ can be obtained by $g_3=g_1\times g_2$. But I…
J.Dillinger
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Finite-order elements of $\text{GL}_4(\mathbb{Q})$

I'm currently studying for my qualifying exams in algebra, and I have not been able to solve the following problem: Determine all possible positive integers $n$ such that there exists an element in $\text{GL}_4(\mathbb{Q})$ of order $n$. I've been…
Debbie
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Why is the quotient map $SL_n(\mathbb{Z})$ to $SL_n(\mathbb{Z}/p\mathbb Z)$ is surjective?

Recall that $SL_n(\mathbb{Z})$ is the special linear group, $n\geq 2$, and let $q\geq 2$ be any integer. We have a natural quotient map $$\pi: SL_n(\mathbb{Z})\to SL_n(\mathbb{Z}/q).$$ I remember that this map is surjective (is it correct?). It…
ougao
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$GL_n(\mathbb F_q)$ has an element of order $q^n-1$

For fixed prime power $q$ show that the general linear group $GL_n(\mathbb F_q)$ of invertible matrices with entries in the finite field $\mathbb F_q$ has an element of order $q^n-1$. I tried to show this question with showing diagonal matrix but…
bytrz
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How is $\mathrm{PGL}(V)$ a subgroup of $\mathrm{P\Gamma L}(V)$?

I've stumbled upon a strange exercise while reading "Notes on Infinite Permutation Groups" by Bhattacharjee, Möller, Macpherson and Neumann. If you have the book, the exercise is 7(ix) on page 66. Let me start with the definitions. Let $F$ be an…
Dan Shved
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Techniques for showing that a subgroup is not normal

To show that a subgroup of a group is normal, I typically construct a homomorphism whose kernel is that subgroup. Are there any general principles or tests that I can use to determine that a subgroup is not normal? Or are such claims usually proved…
Bachmaninoff
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$PGL(n, F)=PSL(n, F)$

$PGL(n, F)$ and $PSL(n, F)$ are isomorphic if and only if every element of $F$ has an $n$th root in $F$. ($F$ is a finite field.) I can show that if $PGL(n, F)=PSL(n, F)$ then $|F|$ have to be even. I have not any idea how to deal with it. Any…
Bobby
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Two dimensional complex group representations

Michael Artin's Algebra, chapter 10 (both unstarred, and complex representations) M.8 Prove that a finite simple group that is not of prime order has no nontrivial representation of dimension 2. M.14 Let $\rho\colon G\to GL(V)$ be a two-dimensional…
Yai0Phah
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Geometrical interpretation of a group action of $SU_2$ on $\mathbb S^3$

Background There're some nomenclatures from Michael Artin's Algebra to explain. 3-Sphere, or $\mathbb S^3$, is the locus of $x_0^2+x_1^2+x_2^2+x_3^2=1$, where $(x_0,x_1,x_2,x_3)\in\mathbb R^4$. $SU_2$ is a special unitary group, i.e, the group of…
Yai0Phah
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Automorphisms of $GL_2(\mathbb{Z}/p^{n}\mathbb{Z})$

Let $p$ a prime and $n$ a positive integer, What are the outer automorphisms of the finite linear group $GL_2(\mathbb{Z}/p^{n}\mathbb{Z})$? do we know a complete list of them? Is there any thing on the literature about this? After some search in…
Rachid
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Does $G$ being a subgroup of $GL(n, \mathbb Z/p\mathbb Z)$ for all odd primes $p$ imply it is a subgroup of $GL(n, \mathbb{Z})$?

It is known that any finite subgroup of $GL(n, \mathbb{Z})$ is isomorphic to a subgroup of $GL(n, \mathbb Z/p\mathbb Z)$ for any odd prime $p$ (see here). I am wondering if there is a converse to to this: Does $G$ being isomorphic to a subgroup of…
Beren Gunsolus
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Is this group of matrices cyclic?

Is the group $H$ consisting of matrices of the form $ \left( {\begin{array}{cc} 1 & n \\ 0 & 1 \\ \end{array} } \right) $ cyclic, where $n \in \mathbb{Z}$? If not, how would you show this?
M Smith
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Abelianization of general linear group?

I am asking purely out of interest: What the abelianization of general linear group $GL(n,\mathbb{R})$?
Zuriel
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Conjugacy classes in $SU_2$

I'm trying to find all conjugacy classes in $SU_2$. Matrices in $SU_2$ are of the form: $M = \begin{bmatrix} \alpha & \beta \\ - \bar{\beta} & \bar{\alpha} \end{bmatrix}$ and $|\alpha|^2+|\beta|^2 =1$ I thought I could use…
Don
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