4
votes
1answer
55 views

Order of group $GL_{2}\left( \mathbb{F}_{p}\right) $

I'm having a hard time counting. I need to count the number of elements for the multiplicative group of invertible $2\times 2$ matrices $GL_{2}\left( \mathbb{F}_{p}\right) $ with elements from the ...
17
votes
2answers
193 views

Show that the kernel of the map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion.

I am trying to show that the kernel of the natural map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion. That is, if $A$ is in the kernel then $A = I$ or $A^n \neq I$ for all $n ...
2
votes
1answer
16 views

Eigenvectors of a Lie group invariant covariant matrix

Suppose you have a $n\times n$ covariance matrix $C$ that is commuting with all group elements, $g$, of a non abelian Lie group $G$, i.e. $[C,g]=0$ for all $g \in G$. Can we derive explicitly the form ...
2
votes
1answer
86 views

Projective special linear groups are co-hopfian?

Is it known if $PSL(2,\, F)$, with $F$ a field of prime $p$ characteristic (maybe with all proper subfields of finite order), is co-hopfian? I've searched everywhere but found nothing. Definition A ...
5
votes
1answer
54 views

Is S a group under matrix addition

Another matrix question! Let $$S=\{A \in M_2(\mathbb{R}):f(A)=0\}\text{ and }f\left(\begin{bmatrix}a&b\\c&d \end{bmatrix}\right)=b$$ Is S a group under matrix addition. Either prove that ...
2
votes
1answer
52 views

If direct limits of matrices are isomorphic, is the direct limit of the transpose matrices also isomorphic?

On the one hand, the following conjecture seems reasonable, but on the other hand it doesn't seem natural because some objects are being dualised while others are not. I would appreciate if anyone ...
9
votes
2answers
109 views

Subgroups of $\mathrm{GL}(n,\mathbb{Z})$ which are not finitely generated

The group $\mathrm{GL}(n,\mathbb{Z})$ is finitely generated: take for example diagonal matrices, permutations and one elementary matrix (upper triangular). Are there some simple / nice examples of ...
12
votes
1answer
108 views

Check membership in a matrix group

I'm looking for a (preferably somewhat efficient) algorithm for this problem: Given a normal subgroup of $SL(m, \mathbb{Z})$ generated by a finite set $\{M_1, M_2, \dotsc, M_n\}$, and some $A \in ...
2
votes
1answer
41 views

Isomorphism between groups of $2 \times 2$ matrices

I'm stuck on this problem: For $\mu \in \mathbb{R} \setminus \{1\}$ let $$G_\mu := \left\{\begin{pmatrix}a & b \\ 0 & a^\mu \end{pmatrix} : a \in \mathbb{R}^+, \; b \in \mathbb{R}\right\} .$$ ...
1
vote
2answers
59 views

Conjugates of the upper triangular matices

It's a shame...I want to give an explicit description of the set $\bigcap_{m \in GL_n(K)} mUm^{-1}$, $U$ being the upper triangular subgroup of $GL_n(K)$. It seems to be $K^\times I_n$ but I do not ...
0
votes
1answer
11 views

Forms - unitary group?

If $Id$ is the $n$ by $n$ identity matrix and $J$ is the $2n$ by $2n$ matrix with $Id$ in the upper right corner and $-Id$ in the lower left corner, then $Sp_{2n} = \{G\in Gl_{2n} : G^{tr}JG = J\}$ is ...
4
votes
2answers
121 views

Do cyclic permutations of rows and column entries generate all permutations?

Background: I am interested in the group of permutations of the entries of a general $m\times n$ matrix. In particular, I am interested in (1) interesting sets of simple generators for this group ...
3
votes
3answers
177 views

Do row and column permutations generate all permutations?

Suppose $m,n\ge 1$ are integers. Do row and column permutations of an $m\times n$ matrix generate the group of all permutations of the $mn$ entries of the matrix? More formally, let $A_1$ be the ...
3
votes
1answer
56 views

the representation of a free group

A group $G$ is generated by $\begin{pmatrix}1&n\\0&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\n&1\end{pmatrix}$, then we know $G\cong \mathbb{F}_2$ which is a free group generated by two ...
2
votes
2answers
66 views

Naive question about the group $SU(n)$?

As usual, let $SU(n)$ represent the set of all the $n\times n$ unitary matrices with determinant $1$. It's easy to show that any matrix $U$ takes the form $U=e^{iA}$ ($A$ is a $n\times n$ traceless ...
3
votes
0answers
48 views

Continuous subgroup of SO(3)?

I read from a paperarXiv: cond-mat/0602109 by a theoretical physicist, Prof. Frank Bais, close subgroups of $SO(3)$ is given by ${C_n,D_n,T,O,I,SO(2)\rtimes Z_2}$, where $C_n$ is the cyclic group of ...
1
vote
1answer
35 views

Subgroups of GL(2,C) isomorphic to Z

Let $\mathbb Z\to \mathrm{GL}_2(\mathbb C)$ be an injective homomorphism. I'm wondering about the possibilities for the image of $\mathbb Z$. I think the image is always conjugate to a subgroup of ...
1
vote
1answer
42 views

get an element by finitely generated set

I want to know the method to get a element in a finitely generated group by its generated set, is there a general way to calculate? For example, $SL(2,\mathbb{Z})=<a,b|a=\begin{pmatrix}0 &1\\-1 ...
4
votes
2answers
82 views

The periodic nature of the fibonacci sequence modulo $m$

Let $x_n$ denote the $n$-th element of the fibonacci sequence and $$A:=\begin{pmatrix} 0&1\\1&1 \end{pmatrix}$$ It's easy to show, that it holds: $$A^n=\begin{pmatrix} ...
3
votes
1answer
62 views

What is the maximal torus in the Lorentz group $O(m,n)$?

I'm close to certain it's just the product of the maximal tori of $O(m)$ and $O(n)$, but I can't quite prove it. I've tried the following: ...
2
votes
1answer
33 views

Matrix Groups in Abstract Algebra

QUESTION: Let $h= \begin{pmatrix} -1 & 1\\-1&0 \end{pmatrix} \in GL_2(\Bbb R)$. Find $\langle h\rangle$. I'm stuck on the solution, but here is what I have: Let $h=\begin{pmatrix} ...
8
votes
2answers
106 views

Order of invertible matrices

I recently came across an interesting problem in Artin which says: If $A \in GL_2(\mathbb{Z})$ is of finite order then it has order $1,2,3,4,6$. I was looking for a generalization of this problem. ...
1
vote
1answer
43 views

Existence of isomorphism between groups of upper triangular matrices.

Is there an isomorphism between this group of matrices $$ \begin{pmatrix} 1 & k \\ 0 & 1 \\ \end{pmatrix},~~k\in\mathbb Z $$ and this one $$ \begin{pmatrix} 1 & k_1 & k_2 \\ 0 & 1 ...
4
votes
1answer
150 views

Abelian group generators and relations

(a) Define what it means for an abelian group to be finitely generated. Explain the terms elementary divisors and rank of $G$ and describe the structure theorem for finitely generated abelian ...
0
votes
1answer
55 views

Prove that the group of $3\times3$ rational unipotent triangular matrices modulo its center is isomorphic to the additive group $\mathbb Q^2$

Let $G$ be the group of matrices of the form: $$\left( \begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \\ \end{array} \right)$$ with $a,b,c \in ...
0
votes
1answer
24 views

suggest me a example for non singular, conjugate-symmetric sesquilinear form ????

I only know that fact that the matrix corresponding to the non singular, conjugate-symmetric sesquilinear form is a unitary matrix. and SU_n(q) is the unitary groups is the collection of the ...
2
votes
1answer
54 views

Classification of parabolic elements of a subgroup of $PSL_2(\mathbb R)$

Let $G\subset PSL_2(\mathbb R)$ be the group generated by the matrices $$a_n=\begin{pmatrix} 1 & 2\cot\frac{\pi}{n}\\0 & 1\end{pmatrix},\; c_n = \begin{pmatrix} ...
2
votes
1answer
96 views

Abstract Algebra Matrix Group Theory

The matrix group G = SL(n, $\mathbb{R}) = \{A \in M(n, \mathbb{R})\} \text{ acts on } X= R^n$ by left matrix multiplication: $\tau _A(x) = A\cdot x (\text{matrix product }(n \times n ) \cdot (n \times ...
1
vote
1answer
45 views

prove that the maximal torus of $SO(3)$ is the maximal torus of $GL_3(\mathbb{R})$.

I want to prove that the maximal torus of $SO(3)$ is the maximal torus of $GL_3(\mathbb{R})$. I want to use the theorem that every maximal torus of G equals $gTg^{-1}$ for some $g \in G$. But I am not ...
2
votes
1answer
79 views

Determining the structure of the abelian group, integral matrix

I am revising for my upcoming university exams and I have a past exam question that I am finding particularly challenging... a) Consider the integral matrix $$R=\begin{bmatrix} 2 & 2 & ...
4
votes
1answer
48 views

Closed conjugacy classes in $M_n(k)$

Let $k$ be an algebraically closed field, $n$ a positive integer, and consider the action of $\mathrm{GL}_n(k)$ on $M_n(k)$ by conjugation. My professor tells me that semisimple conjugacy classes are ...
2
votes
1answer
46 views

Upper Unitriangular Matrices

Let $U$ be the group of the upper unitriangular matrices $n$-$n$ over the field of rationals $\mathbb{Q}$. I know that $U$ is nilpotent and torsion-free. It is also radicable? How it can be proved in ...
1
vote
0answers
35 views

Is a matrix a subgroup of a group when its the inverse matrix “looks different”?

I have the to prove whether a subset of a group is a subgroup. The following subset is given: $$U = \left\{ \begin{pmatrix} a & b & 0 \\ 0 & 1 & c \\ 0 & 0 & d \end{pmatrix} ...
1
vote
1answer
22 views

Proof, wheather a subset of a Group is a Subgroup

I have to check, weather the following subset of a group is also a subgroup: $$U = \left\{ \begin{pmatrix} a & -b \\ \overline{b} & \overline{a} \end{pmatrix} \in GL(2, \mathbb{C}) \bigg\vert ...
7
votes
1answer
58 views

Orbits of action of $SL_m(\mathbb{Z})$ on $\mathbb{Z}^m$

I'm considering the action of $SL_m(\mathbb{Z})$ on $\mathbb{Z}^m$: if $A\in SL_m(\mathbb{Z})$ and $v\in\mathbb{Z}^m$, then $Av\in\mathbb{Z}^m$. My question is: what are the orbits of this action? ...
0
votes
1answer
50 views

Show that matrix under addition is isomorphic with the group of complex numbers under addition

Q: Let $M = \{ \begin{bmatrix} a & b \\ -b & a \end{bmatrix} | a, b \in \mathbb R \}$. Show that $(M,+)$ and $(\mathbb C,+)$ are isomorphic. Show that $(M^{*},*)$ and $(\mathbb C^{*},*)$ are ...
1
vote
1answer
62 views

What mathematical algebra explains sequence of circular shifts on rows and columns of a matrix?

Consider a simple matrix (3X3) with entries thus: [1 2 3; 4 5 6; 7 8 9;] Circular shifts can be performed on any row or any column thus: row-(1/2/3)-(right/left) and column-(1/2/3)-(up/dn) ...
2
votes
1answer
36 views

center of invertible matrices

find the center of the group of invertible 2 x 2 matrices with real entries. Attempt: By definition, the center of a group Z(G), is where all the elements are commutative. If G = { invertible 2 x 2 ...
0
votes
0answers
36 views

Ideals in the unitary group

What would be examples of one-dimensional ideals in the lie algebra of the unitary group? Moreover, how would one show that it is in the tangent space of the center of the unitary group and that the ...
3
votes
0answers
42 views

Isomorphism Classes

I'm trying to find what the isomorphism class of the group of $G = n\times n$ matrices with $\pm 1$ along the diagonal and zeroes everywhere else. My approach was first to show that because for each ...
4
votes
1answer
85 views

Commutator subgroup of $GL_{2}(\mathbb{Z}/p^{2}\mathbb{Z})$ is $SL_{2}(\mathbb{Z}/p^{2}\mathbb{Z})$

How would I go about showing this, where $p$ is an odd prime? The inclusion $[GL_{2}(\mathbb{Z}/p^{2}\mathbb{Z}),GL_{2}(\mathbb{Z}/p^{2}\mathbb{Z})] \subseteq SL_{2}(\mathbb{Z}/p^{2}\mathbb{Z})$ is ...
1
vote
1answer
54 views

Proving group properties of $G$, a set of $2 \times 2$ matrices with rational entries

Let $G$ be the set of all $2 \times 2$ matrices whose entries are rational numbers and whose determinant is equal to $3^n$ where $n$ is a nonnegative integer. Prove that $G$ is a group with respect ...
0
votes
1answer
41 views

What is the order of this group? [duplicate]

Let $H$ be the subgroup of the group $G$ of all $2 \times 2$ non-singular matrices whose entries are integers modulo a given prime $p$ consisting of those and only those matrices in $G$ whose ...
0
votes
1answer
76 views

Find the inverse of a matrix in $GL(2\,,\, \Bbb Z_{11})$.

What are the necessary steps and reasoning for calculating the following matrix in GL(2,$\Bbb Z_{11}$): $M = \begin{pmatrix} 2&6 \\3&5 \end{pmatrix}$. I found the answer to be ...
0
votes
1answer
60 views

How to prove that $J(M_n(R))=M_n(J(R))$?

How to prove that $J(M_n(R))=M_n(J(R))$? Here $M_n(R)$ is the ring of matrices of size $n^2$ over the ring $R$. And $J(M_n(R))$ is a two-sided ideal of the ring $M_n(R)$.
0
votes
0answers
70 views

Show that $H$ is a normal subgroup of $G$? [duplicate]

Let $\mathbb M(n;\mathbb R)$ denote the set of all real matrices (identified with $\mathbb R^{n^2}$ and endowed with its usual topology) and $GL(n;\mathbb R)$ denote the group of all invertible ...
2
votes
2answers
49 views

subgroup of $GL_{n+1}\mathbb{R}$ which is isomorphic

Describe a subgroup of $GL_{n+1}\mathbb{R}$ which is isomorphic to the group $\mathbb{R}^n$ under the operation of vector addition. I have no idea what this would look like. I would really ...
2
votes
3answers
48 views

Quaternion identity proof

If $q \in \mathbb{H}$ satisfies $qi = iq$, prove that $q \in \mathbb{C}$ This seems kinda of intuitive since quaternions extend the complex numbers. I am thinking that $q=i$ because i know that $ij = ...
1
vote
1answer
40 views

Matrix Groups dealing with $GL_n(\mathbb{R})$

Describe all elements $A \in GL_n(\mathbb{R})$ with the property that $AB = BA$ for all $B \in GL_n(\mathbb{R})$. I would really appreciate any help. I am not sure where to start. I almost want to ...
2
votes
0answers
45 views

Conjugacy classes of unipotent $\mathbb{Z}\times\mathbb{Z}$ in $GL_3(\mathbb{Q})$

Let $G=\mathrm{GL}_3(\mathbb{Q})$. Now, consider all subgroups in $G$ of the form $\mathbb{Z}\times\mathbb{Z}$ consisting only of unipotent elements (elements whose eigenvalues are all $1$). How ...