Questions tagged [general-linear-group]
The general linear group of order $n$, $GL_n(\mathbb{F})$, over a field $\mathbb{F}$ (usually $\mathbb{R}$ or $\mathbb{C}$) is the group of $n\times n$ invertible matrices over $\mathbb{F}$. The operation is the usual matrix multiplicatoin.
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Finding LFSR periods
We have LFSR with corresponding linear automata $A \in F_2^{n\times n}, B \in F_2^n$. We know that this LFSR's generating function is $B^T * (A - I_n * x)^{-1} * S$, where $S$ is the state from $F_2^n$...
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Equation in $GL(2,\mathbb{Z}_{2^{n-1}})$
I'm now studying how to embed generalized quaternion group $Q_{2^n}=a^{2^{n-1}}=e$, $u^2=a^{2^{n-2}}$, $ua=a^{-1}u$ in $GL(k,\mathbb{Z}_{2^{n-1}}).$ I got the embedding in case $k=3$, but $k=2$ is ...
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$GL_n(C)$ is isomorphic to a lie subgroup of $GL(2n,R)$
$GL_n(C)$ is isomorphic to a lie subgroup of $GL_{2n}(R)$.
I see some posts concerning this (not the same claim):
$GL(n, \mathbb{C})$ is isomorphic to a subgroup of $GL(2n, \mathbb{R})$
$GL_n\mathbb{C}...
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A non-trivial homomorphism from $SL(2, 7)$ to $GL(4, 11)$.
Let $G=GL(4, 11)$ be the general linear group of $4\times 4$ matrices over the field $\mathbb{F}_{11}$. Let $H=SL(2, 7)$ be a special linear group of $2\times 2$ matrices over the field $\mathbb{F}_{...
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Generating set of General linear group [closed]
What is one possible minimal generating set of the general linear group $GL_{m}(Z_{p})$?
It might be very easy question whose solution is known to everyone except me. Kindly help me with the same.
...
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No automorphisms of order $p^2$
Let $H$ be the group of integers mod p, under addition, where
$p$ is a prime number. Suppose that n is an integer satisfying $1 ≤ n ≤ p$,
and let G be the group $H × H × · · · ×H $($n$ factors). Show ...
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Prove that Aut($\mathbb Z \times \mathbb Z$) $\cong$ $\text{GL}_2(\mathbb Z)$
Prove that Aut($\mathbb Z \times \mathbb Z$) $\cong$ $\text{GL}_2(\mathbb Z)$.
This is a HW problem for an Algebra course, hints/suggestions welcome.
I didn't find this problem on math.SE, however I ...
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Polar decomposition on $\mathbf{GL}(n,\mathbb{R})$
I am asked to prove that for $g\in\mathbf{GL}(n,\mathbb{R})$, $g$ can be unique written as $$g=k_1
\begin{pmatrix}
a_1 & & 0\\
& \ddots &\\
0 & & a_n
\end{...
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Proof techniques to show a representation is faithful
I am curious what proof method is most commonly used to show that a representation is faithful. I have found remarkably little online about this question..
It makes sense how to show that a ...
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Chapter 1, Exercise 12 in Brian Hall's Lie groups, Lie algebras, and representations
Suppose $A$ and $B$ are invertible $n \times n$ matrices. Show that there are only finitely many complex
numbers $\lambda$ for which $\text{det}( \lambda A + (1-\lambda)B) = 0$. Show that there exists ...
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Why $\mathrm{SL}$-invariants and highest weight vectors of rectangular shape coincide?
Let $V = (\mathbb{C}^n)^{\otimes d}$ with action of group $G = \mathrm{GL}(n)^{\times d}$ acting componentwise. Consider two spaces:
$A = \mathbb{C}[V]^{\mathrm{SL}(n)^{\times d}}_{nk}$, the space of ...
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Embedding the generalised quaternion group into a general linear group
It's known that there are four non-abelian groups with cyclic subgroup of index $2$. Those groups are the dihedral group $D_{2^n}$, generalised quaternion group $Q_{2^n}$, modular-maximal group $M_{2^...
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Action of $GL(2,\mathbb{Z})$ on lower half plane of $\mathbb{C}$?
I began reading about modular forms and I had a question. So, I know that $SL(2,\mathbb{Z})$ is mapped to the upper half complex plane using a function. The way I see it, the reason we map it to the ...
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Constructive proof of: The general linear group over $\mathbb{F}_2$ is perfect for $n \geq 3$
Any $A \in GL_n (\mathbb{F}_2)$ can be written as the group commutator $A = DBD^{-1}B^{-1}$ with $D,B \in GL_n (\mathbb{F}_2)$ if $n \geq 3$.
In Theorem 1 of R. C. Thompson's thesis the statement ...
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$S_4$ as a subgroup of $GL_3(\mathbb{F}_2)$
I am trying to find a subgroup of $GL_3(\mathbb{F}_2)$, which is isomorphic to $S_4$. Our teacher gave us a hint: we should look at matrices with first column $(1, 0, 0)^T$.
But what is the next step?
...
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General linear group topological structure
(a) Prove that the exponential map defines a bijection between the set of all Hermitian
matrices and the set of positive definite Hermitian matrices.
(b) Describe the topological structure of $GL_2(\...
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$GL(n, \mathbb{C})$ is not residually finite
I saw in some text that $GL(n,\mathbb{C})$ is not residually finite since it is connected. I am not sure about this implication. I guess the proof works somehow like this: we endow $GL(n, \mathbb{C})$ ...
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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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What's the maximum order of an element in $SL_2(\mathbb{Z} /p\mathbb{Z})$ for $p>2$ prime?
I know the answer is $2p$ as I've checked it for $p=3,5$ and $71$.
The characteristic polynomial of a matrix $A\in$ $SL_2(\mathbb{Z} /p\mathbb{Z})$ is $P_A(x)=x^2-tr(A)x+1$, so if this polynomial has ...
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Finding $|Aut(\mathbb{Z}_6 \times \mathbb{Z}_{10} \times \mathbb{Z}_{15})|$ [duplicate]
I am trying to compute:
$|Aut(\mathbb{Z}_6 \times \mathbb{Z}_{10} \times \mathbb{Z}_{15})|$
I know that:
If $H$ and $K$ have coprime orders then:
$Aut(K \times H) \cong Aut(K)\times Aut(H) $
For ...
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Linear group contained in symplectic group
Do we have SL$(n,2) \leqslant$ Sp$(2n, 2)$? And better yet, $\operatorname{SL}(n,q) \leqslant$ Sp$(2n, q)$ or
$\operatorname{GL}(n,q) \leqslant$ Sp$(2n, q)$?
I checked using Magma for small degrees ...
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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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Why are linear algebraic groups defined the way they are? What is the intuition?
Definitions
A subgroup $G \subset GL(n, \mathbb{C})$ is a linear algebraic group, if there is a set $A$ of
polynomial functions on $Mn(\mathbb{C})$ such that $G = \{g \in GL(n, \mathbb{C})| f(g) = 0 \...
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How does go from $GL(V)$ to $GL_n(F)$? What is the relationship between these groups? [duplicate]
How to go from $GL(V)$ to $GL_n(F)$?
During my exams, I was asked what is the difference between $GL(V)$, $GL_n(F)$, $GL_n(R)$, $GL_n(\mathbb{R})$ etc. and how could one go for example from $GL(V)$ to ...
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Conjugacy in $\text{GL}(2, \mathbb{Z})$ vs $\text{GL}(2, \mathbb{Q})$
When are two elements $x,y\in\text{GL}(2, \mathbb{Z})$ conjugate in $\text{GL}(2, \mathbb{Q})$, but not in $\text{GL}(2, \mathbb{Z})$? Does this ever happen? I feel that it should sometimes be the ...
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Are a $2\times 2$ matrix in $\text{SL}(2,\mathbb{Z})$ and its transpose conjugate in $\text{GL}(2,\mathbb{Z})$?
I've been studying some math by myself this summer, and have recently been doing some reading about the groups $\text{GL}(2,\mathbb{Z})$, $\text{SL}(2,\mathbb{Z})$, etc. I've been trying to get a ...
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Maps from a profinite group to $\operatorname{GL}_n(\mathbb R)$.
Suppose $G$ is a profinite group (a topological group which is compact, Hausdorff and totally disconnected). Let $r:G \to \operatorname{GL}_n(\mathbb R)$ be a group homomorphism.
How to prove: $r$ is ...
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About the faithfulness of cuspidal characters
Assuming that we are working over the field of complex numbers, let x be a cuspidal character of the general linear group $GL_n(F)$ over the finite field F. Can we say that x is always faithful? For n ...
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Is there a general linear group version of general relativity (not diffeomorphism)?
General relativity is strongly based on the concept of diffeomorphism.
However diffeomorphisms are not linear.
I wish to recover as much of general relativity as possible, but while remain linear.
I ...
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Finding maximal, solvable, primitive subgroups of a large group in GAP
I am trying to find the maximal, solvable, primitive subgroups of a large group $N$ which is itself a subgroup of $GL(n,p)$ for $(n,p)=(4,3),(4,5),(4,7),(6,3),(10,3)$. However, GAP is too slow to run ...
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A Problem of Linear Algebra that can Stumps Anyone
This problem encountered me during the study of balanced Boolean functions. I was in need of constructing those balanced Boolean functions for which $\mathcal{D}_bf\ne 1$. And finding the solution to ...
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Abelianization of $PGL(n,F)$
For any field $F$, it's known that the abelianization of $GL(n,F)$ is $F^{*}$ except for $GL(2,\mathbb{F}_{2})$. What is the abelianization of $PGL(n,F)$? It seems to me there are non-trivial elements ...
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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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Correspondence closed subgroup of $GL(n,k)$ and closed subgroup scheme of $GL_n$
Sorry for my bad English.
Let $k$ be a field (if necessary ch $k=0$).
We can think a general linear group $GL(n,k)$ is a topology group by Zariski topology, and hence think closed subgroups $H\subset ...
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Find all subgroups of $\mathrm{GL}(2,\mathbf{R})$ of index 2.
Question: We want to find all subgroups of $\mathrm{GL}(2,\mathbf{R})$ of index $2$.
Here is my first attempt: We already know that for a group $G$, if a subgroup $H$ satisfies $(G\colon H)=2$, then $...
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Existence of a transvection which maps one hyperplane to another.
Following screenshot is of a proof from the book Classical groups and geometric algebra by Larry C. Groove. I have some doubt in the proof. My question:
How $\tau$ becomes a transvection on $V$?
To ...
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Existence of $n\times n$ invertible matrix of order $n+1$
Let $K$ be a field of characteristic distinct to $2$ and $n$ be a natural. I want to know if my proof of the existence of $A\in \text{GL}_n(K)$ of order $n+1$ is correct:
Consider $A$ to be the ...
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When is it possible to find an invertible matrix of order $n+1$
Let $n$ be a natural and $K$ a field. When can we find an $n\times n$ invertible matrix $A$ of order $n+1$? Order here means $A^{n+1}=I$ and is the smallest integer that satisfies it.
This is what I ...
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Matrix and Groups Question Solution Check [duplicate]
Group $G$ is the set of all matrices that are real and invertible with this form: $\begin{pmatrix}m&-n\\ n&m\end{pmatrix}$
Show that $f : C^∗ → G$ defined by $f(m + ni) = \begin{pmatrix}m&-...
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$GL_n(\mathbb{Z}_{pq})$ isomorphic to $GL_n(\mathbb{Z}_{p})\times GL_n(\mathbb{Z}_{q})$
I was trying to solve the problem of finding the number of elements of $GL_n(\mathbb{Z}_m)$ for some $m\in\mathbb{N}$. The case where $m=p$ prime was easy because of elementary linear algebra. The ...
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Show that the general linear group $GL(n, \Bbb R)$ is a Lie group under matrix multiplication.
Show that the general linear group $GL(n, \Bbb R)$ is a Lie group under matrix multiplication.
I'm reading an introduction to manifolds by Tu and found this problem there. The definition I have for a ...
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Orbits of the action of $SL(n,\mathbb R)$ on $\mathbb R^n$.
We know that $GL(n,\mathbb R)$ acts on $\mathbb R^n$ via left multiplication. We can easily see that there are two orbits viz $\{0\}$ and $\mathbb R^n-\{0\}$. Now we also know that if $G$ acts on $X$ ...
- 2,957
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Order of completely reducible group action
Let $\mathbb{K}$ be an algebraically closed field and $V$ an n-dimensional $\mathbb{K}-$vector space. Suppose $G \leq GL(V)$ is completely reducible and that for some $d \in \mathbb{N}$ we have $ g^d =...
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Finding the order of an element in $GL(2,\mathbb F_p)$.
Let $\mathbb F_p=\mathbb{Z}/p\mathbb{Z}$. I am looking for a general method of finding the order of an element in $GL(2,\mathbb F_p)$. Suppose $p=7$ and I am given the element $\begin{pmatrix} 5 &...
- 2,957
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Explicit construction of polynomial representation of $GL(n,\mathbb{R})$
In order to understand representations of the general linear group $GL(n,K)$, with $K=\mathbb{R}$ or $K=\mathbb{C}$,
I'm looking for an explicit construction of the polynomial representation matrices
...
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Maximal order with primitive determinant in $\operatorname{GL}_n(\mathbb{F}_q)$
The following question has come up in a facet of a current project. Having an answer (hopefully affirmative) will help me design and test some computational simulations.
$\mathbb{F}_q$ denotes the ...
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When is the set of all upper triangular matrices in $\text{GL}(2, \mathbb{Z}/p \mathbb{Z})$ is abelian?
In page 4 of this paper, it has been mentioned that the collection
$$B=\left\{\begin{pmatrix} a & b \\0 & c\end{pmatrix}~:~a,c \in (\mathbb{Z}/4 \mathbb{Z})^{\times},~ b \equiv 0~(\text{mod}~2)...
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p-subgroup of $\mathrm{GL}_{n}(\mathbb{F }_{p})$ and Sylow's Theorem
Consider the Sylow p-subgroup of $\mathrm{GL}_{n}(\mathbb{F%
}_{p})$ by taking the group of upper triangular matrices with ones along the
diagonal, namely $UT(n,\mathbb{F}_{p})$. $\mathrm{GL}_{n}(\...
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Defining a group in GAP without having its presentation
I know how to define a group when we have its presentation in GAP by using FreeGroup command over the generators and then taking quotient over the relators, however what if I only have the group as a ...
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2
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Find infinitely many homomorphism from $GL_2(\mathbb Q)$ to $\mathbb Q^*$
Find infinitely many homomorphism from $GL_2(\mathbb Q)$ to $\mathbb Q^*$. Here $\mathbb Q^*$ means the multiplicative group of nonzero rational numbers.
My attempt: An example is the determinant ...
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