All Questions
Tagged with matrix-groups or linear-groups
302 questions
2
votes
0
answers
82
views
Topological proof that $\operatorname{PGL}_n$ is embedded in $\operatorname{GL}_{n^2}$?
Summary of the question:
I wanted to know if there is a "simple" topological argument to prove that the inclusion
$\operatorname{PGL}_n \rightarrow \operatorname{GL}_{n^2}$ is a topological ...
- 21
4
votes
0
answers
81
views
Inner automorphism of finite dimensional algebra
There is a claim on Baker's Matrix Groups about inner automorphisms which states the following:
Proposition 4.49: Let $A$ be a finite dimensional (normed) algebra
over $\mathbb{R}$. Then the inner ...
- 353
4
votes
1
answer
313
views
Interesting antihomomorphisms of matrices?
Matrices have two obvious and basic antihomomorphisms (maps satsisfying $\phi(AB)=\phi(B)\phi(A)$):
Transposition: $(AB)^T=B^TA^T$
Inversion: $(AB)^{-1}=B^{-1}A^{-1}$ (where here we consider only ...
4
votes
2
answers
140
views
References on the general linear group over a finite field (finite general linear group) $GL_n(\mathbb{F}_q)$
Does anyone know any good references on the structure of/general facts about $GL_n(\mathbb{F}_q)$ which is to say the general linear group over a finite field of order q (with $q=p^k$ for p a prime)? ...
0
votes
0
answers
52
views
A profinite presentation of ${\rm SL}_{2}(\mathbb{Z}_{p})$
Let $\mathbb{Z}_{p}$ be the ring of $p$-adic integers and let ${\rm SL}_{2}(\mathbb{Z}_{p})$ be the two dimensional special linear group over $\mathbb{Z}_{p}$. Note that ${\rm SL}_{2}(\mathbb{Z}_{p})$ ...
- 627
1
vote
1
answer
119
views
Representation theory of the affine group - reference request.
I'm currently working on an applied problem that boils down to knowing about the representation theory of the affine group $Aff(\mathbb{R}^n)\cong \mathbb{R}^n\rtimes GL(n,\mathbb{R})$. I am ...
- 312
0
votes
0
answers
48
views
Approach to showing that the group of invertible upper-triangular matrices over a finite field is soluble [duplicate]
Let $F$ be a finite field and let $G$ be the subgroup of upper triangular matrices in $GL_n(F)$.
I am working through an exercise to show that $G$ is a soluble group.
The exercise has the following ...
- 999
0
votes
1
answer
86
views
Cardinal longitudes of $SU_2$
I am working out of Artin's Algebra. I was wondering what he meant by "we haven't run across the longitude $cI + s\mathbf{k}$ before." There doesn't seem to be any further mention of this ...
- 45
0
votes
1
answer
76
views
Irreducibility of the symplectic group $\operatorname {Sp}(2n,k)$ as a linear algebraic group
Let $k$ be an algebraically closed field and $G:= \operatorname {Sp}(2n,k)$ the symplectic group.
It is not that difficult to show, that it is in fact a linear algebraic group in both: $k^{n^2+1}$ and ...
- 1,390
0
votes
1
answer
102
views
Center of $SL_2(\mathbb{F}_3)$.
The center of $SL_2(\mathbb{F}_3)$ and $GL_2(\mathbb{F}_3)$ are both $\{I,-I\}$.
In the upper central series of $G=GL_2(\mathbb{F}_3)$, we thus have $Z_1=Z(G)=\{I,-I\}$.
$Z_2$ is defined by the ...
- 1,022
5
votes
2
answers
168
views
What are the possible Galois groups of solvable equations of order $pq$?
In chapter 14 of Galois Theory by David Cox the author classifies (or at the very least describes in a pretty satisfying way) all possible solvable Galois groups of irreducible polynomials over $\...
- 553
0
votes
1
answer
118
views
Polynomial and rational representations of $GL_n(\mathbb{C})$
Let $G = GL_n(\mathbb{C})$.
A rational representation of $G$ is a group homomorphism $\varphi: G \to GL(V)$, where $V$ is a finite-dimensional vector space over $\mathbb{C}$, and $\varphi$ is given by ...
user969954
0
votes
1
answer
73
views
On the adjoint representation of $GL_n(\mathbb{C})$
I'm studying the adjoint representation of $GL_n(\mathbb{C})$, and I'm encountering some confusion about the denominator that appears in the matrix entries of the adjoint action.
The adjoint ...
user969954
1
vote
0
answers
55
views
Is there a non-unimodular group which is not contained in a unimodular group?
The examples I know of non-unimodular (locally compact second countable) groups arise as subgroups the general linear group (e.g. the affine linear group). I was wondering whether there is an example ...
0
votes
0
answers
102
views
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:
...