All Questions
Tagged with general-linear-group linear-groups
19 questions
0
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47
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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:
...
4
votes
1
answer
129
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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 ...
- 643
3
votes
0
answers
86
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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 $p$....
1
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60
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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 &...
- 159
1
vote
0
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404
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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 ...
- 1,487
2
votes
1
answer
208
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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 ...
- 1,103
2
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0
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169
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Do different naming conventions exist regarding what $PGL(n, k)$ means?
According to Hartshorne, Foundations of Algebraic Geometry, $PGL(2, \mathbb R)$ denotes the group of automorphisms of $\mathbb{RP}^2$, the real projective plane, and is therefore a quotient of $GL(3, \...
- 24.2k
2
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0
answers
292
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What is the difference between $PGL(2, \mathbb R)$ and $SL(3, \mathbb R)$?
Every element $\alpha$ in $PGL(2, \mathbb R)$ is represented by a matrix $A \in GL(3, \mathbb R)$; two such matrices $A_1, A_2 \in GL(3, \mathbb R)$ represent the same element $\alpha$ if and only if $...
- 24.2k
2
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401
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Order of $PSL(n,q)$
Being $F$ the field of order $q$ that linear groups are defined here, there is something I can't understand.
I know that $|SL(n,q)| = \dfrac{|GL(n,q)|}{(q-1)}$, and I know that $PSL(n,q) = SL(n,q)/Z(...
- 515
1
vote
1
answer
143
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Finite abelian subgroups of $GL(2,\mathbb C)$ without Pseudoreflection [closed]
Let $G$ be a finite abelian subgroup of $GL(2,\mathbb C)$ such that $A-I_2$ does not have rank $1$ for all $A\in G.$ Then, is it true that $G$ is cyclic?
- 1,682
0
votes
1
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215
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about Cartan's theorem
theorem : Let $SL(n,\Bbb{C})$ be the group of matrices of complex entries and determinant $1$ then $SL(n,\Bbb{C})$ is a regular submanifold of $GL(n,\Bbb{C})$.
proof : $GL(n,\mathbb{C})$ is a Lie ...
- 4,747
3
votes
2
answers
964
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Why is $SL(n, \mathbb{R})$ the kernel of $\det : GL(n, \mathbb{R}) \mapsto\Bbb R^*$?
The special linear group of invertible matrices is defined as the kernel of the determinant of the map:
$$\det:GL(n,\mathbb{R}) \mapsto \mathbb{R}^*$$
In my mind the kernel of a linear map is the set ...
- 769
4
votes
1
answer
242
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Groups between $\operatorname{GL}_n (R)$ and $\operatorname{SL}_n(R)$?
It occurs to me that $R^\times$ (the group of units of a commutative ring) may have a subgroup, say $S \leqslant R^\times$.
It seems that we could then define the group
$$
\operatorname{GL}_n(R; S) = \...
- 5,859
0
votes
1
answer
1k
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Is $GL_n(\Bbb(R))$ and $SL_n(\Bbb (R)) $ compact?
I am very confused as nothing has been mentioned in the question -the metric .Howevere I do have an idea about compactness. Somebody hints (not the full answer) will be appreciated . I will try to ...
- 1,565
0
votes
2
answers
3k
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$\{A \in GL_2(\mathbb R):|\det(A)|=1\}$ is a normal subgroup of $GL_2(\mathbb R)$
I have to prove that $O_2(\mathbb R)=\{A \in GL_2(\mathbb R):|\det(A)|=1\}$ is a normal subgroup of $GL_2(\mathbb R)$.
I tried to go on with the definition of normal subgroup but I don't really know ...
- 973
4
votes
2
answers
1k
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The center of the group of $n\times n$ upper triangular matrices with a diagonal of ones
Let $\mathbb{F}_{p}$ be a finite field of order $p$ and $H_{n}(\mathbb{F}_{p})$ be the subgroup of
$GL_n(\mathbb{F}_{p})$ of upper triangular matrices with a diagonal
of ones. Note that the center $Z(...
- 338
1
vote
1
answer
421
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The orbit space GL(n,R)/O(n)
If $G= GL(n,\mathbb{R})$ and $H= O(n)$ then why the orbit space
$G/H$ is homeomorphic to the space of all upper triangular
matrices with positive diagonal entries?(Here action of $H$ on $G$ is the ...
-1
votes
2
answers
102
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Is $H$ is a normal subgroup of $G$? Yes/NO [closed]
$G = GL_n(\mathbb{R})$ and $H$ is the subgroup of all matrices in $G$ with positive determinant
Is $H$ is a normal subgroup of $G$?
My attempt : Take G= $ \begin{bmatrix} 2 &0 \\ 0& 1 \...
- 14.8k
0
votes
1
answer
47
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Proving that $Aw\in \langle w\rangle \implies A$ is of the form $\lambda I_n$
If $w=\begin{bmatrix} w_1\\ \vdots\\w_n \end{bmatrix}$ is a vector in $K^n$ for a field $K$ and
$A=
\begin{bmatrix}
\lambda & & \\
& \ddots & a_{ij}\\
& & \lambda
\end{...
- 2,679