# 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 $F_2^n$...
1 vote
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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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### 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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1 vote
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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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### 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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1 vote
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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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### $GL_n(\mathbb{Z}_{pq})$ 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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1 vote
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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 $X$ ...
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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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