Tagged Questions

0
votes
0answers
21 views

Conjugacy classes for su(2)

I am wondering how to calculate the conjugacy classes of the Lie algebra su(2). My guess is that they can be easily evaluated under the similarity transformations but I am not sure it that is all to ...
1
vote
1answer
52 views

Groups of transformations

I tried to find literature and articles about groups of transformations, but mostly what I found is either about groups or about transformations. Can you suggest me literature where groups of ...
2
votes
2answers
38 views

On the eigenvalues of a linear transformation $\tau$ such that $\tau^3 = \mathrm{id}$

I am reading the book on representation theory by Fulton and Harris in GTM. I came across this paragraph. [..] we will start our analysis of an arbitrary representation $W$ of $S_3$ by looking ...
2
votes
0answers
22 views

Extending transvections/generating the symplectic group

The context is showing that the symplectic group is generated by symplectic transvections. At the very bottom of http://www-math.mit.edu/~dav/sympgen.pdf it is stated that any transvection on the ...
6
votes
2answers
66 views

Embeddings of $GL(n-1,q)$ into $GL(n,q)$

Let $n>2$ and $q$ be a prime power. I want to find all the embeddings of $GL(n-1,q)$ into $GL(n,q)$. An embedding I first thought of is as following. Let $V$ be an $n$-dimensional vector space over ...
3
votes
2answers
62 views

Silliness: $\exists~X~\text{s.t.}~AX=B \iff B\in R(L_A)$

So, I am asked to prove that the system of linear equations $AX=B$ has $\color{black}{a~solution}$ if and only if $B\in R(L_A)$. $R$ denotes the "range of" and $L_A$ is left multiplication by $A$. If ...
2
votes
1answer
90 views

A question about basis linear space

See $\mathbb{R}$ as a linear space over $\mathbb{Q}$. $B\subset\mathbb{R}$ is a basis with positive reals. Can $B$ be a group where multiplication is standard?
1
vote
2answers
74 views

Is $\operatorname{GL}(n, \mathbb{R})$ with multiplication a group?

I am looking at an exercise that saying $GL(n,\mathbb{R})$ with multiplication, in other words the nxn matrices with real entries together with multiplication is a group. I wonder the following: do ...
1
vote
0answers
47 views

What to take from representation of $S_d$?

I am reading about group representations, and books I read all contain the representation theory for symmetric groups $S_d$. However none of them presents the material in a friendly way. After reading ...
0
votes
1answer
48 views

Additive maps modulo $1$ - what do they look like?

Recently, some convergence problems I have been considering led me to look at additive maps of the torus (which for me is the additive group $T := \mathbb{R}/\mathbb{Z}$). A map $f:\ T \to T$ is ...
1
vote
0answers
32 views

Computing Invariant Subspaces of Matrix Groups

Does anyone have a program written in Mathematica (or SAGE or GAP) that computes the invariant subspace lattice of a matrix group?
7
votes
1answer
252 views

Do these two sets of matrices form groups?

Stimulated by some Physics backgrounds, consider the following two sets of matrices. Notations and definitions:Let $A,B$ be two complex $n\times n$ matrices, then $\left [ A,B \right ...
2
votes
3answers
115 views

Which group is isomorphic to $\left\langle\begin{bmatrix}0&1\\1&0\end{bmatrix},\begin{bmatrix}1&-1\\0 & -1 \end{bmatrix} \right\rangle$?

Both matrices have determinant equal to -1, so their products are matrices with determinant $\in \{1,-1\}$. Can I conclude that this is isomorphic to $ O_2(\mathbb{R}) $ ?
3
votes
3answers
114 views

Are Clifford groups very *non-commutative*?

Clifford groups seem to be very non-commutative by the relation \begin{equation} \gamma_{i}\gamma_{j}=-\gamma_{j}\gamma_{i}. \end{equation} But is it really so? Can we put this degree of ...
6
votes
2answers
90 views

let$ G=\{M_1,M_2,…,M_k\}$ be a finite group if $\sum _{i=0}^k \operatorname{tr} (M_i)=0$ then how prove $\sum _{i=0}^k M_i=0$

Let $G=\{M_1,M_2,...,M_k\}$ be a finite set such that $ M_i\in M_n(\mathbb R)$ and $(G,\;\cdot\:)$ is group with operations of matrix multiplication If $\sum _{i=1}^k \operatorname{tr} (M_i)=0$ ...
3
votes
0answers
35 views

Duality of $Z(G)$ and $[G,G]$ in representation?

This question and its many wonderful answers illustrate many faces of the duality of $Z(G)$ and $[G,G]$, the centre/ commutator duality of a group. I was thinking about its manifestation in group ...
2
votes
0answers
33 views

Why does $d_{\alpha}$ divide $\#G$ for $\alpha\in\hat{G}$?

Let $\alpha$ be a unitary irreducible representation of a finite group $G$. Then we have \begin{equation} d_{\alpha}|\#G, \end{equation} where $d_\alpha$ is the degree of the representation and $\#G$ ...
7
votes
1answer
73 views

Show $SL(2,\mathbb{Z})$ written as finite product of elements of a particular form

Prove that any element of $SL(2,\mathbb{Z})$ can be represented by a finite product of matrices of the following form. $$\begin{pmatrix}1-ab & a^2\\ -b^2 & 1+ab\end{pmatrix}.$$ We are given ...
2
votes
1answer
94 views

Proof relation between Levi-Civita symbol and Kronecker deltas in Group Theory

In order to proof the following identity: $$\sum_{k}\epsilon_{ijk}\epsilon_{lmk}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$$ Instead of checking this by brute force, Landau writes de product of ...
1
vote
1answer
32 views

Studying the action of $GL(V)$ on the vector space $V$

The statement I am trying to prove is the following. Let $k$ a field and $V$ a $k$-vector space of finite dimension. Let $\mathscr{B}$ be the set of ordered $k$-bases of $V$. The natural ...
2
votes
2answers
95 views

Is $SL_n(\mathbb{R})$ a normal subgroup in $GL_n(\mathbb{R})$?

Let $N= GL_n(\mathbb{R})$ and $M=SL_n(\mathbb{R})$. Is $M$ normal subgroup in $N$? Why or why not? I know how to do this with $GL_2(\mathbb{R})$ and $SL_2(\mathbb{R})$ but with $N= GL_n(\mathbb{R})$ ...
3
votes
1answer
25 views

action of orthogonal group on the space of antisymmetric bilinear forms

What is the natural action of orthogonal group on the space of antisymmetric bilinear forms.
1
vote
2answers
42 views

Property of subgroups

Is the following property correct? Property: let $(G;f_G)$ group, $B \subseteq G$ and $B \neq \emptyset $, then $(B;f_G|_B)$ is group iff $\forall a,b \in B$ we have that $a f_G|_B b'\in B $ and $ b ...
7
votes
1answer
63 views

Finiteness of groups preserving a symmetric positive definite bilinear form

This question arises from reading the note Hodge cycles on abelian varieties by P. Deligne (notes by J.S. Milne). Suppose we are given a group $G$ (for example, either a fundamental group $\pi_1(S, ...
1
vote
1answer
34 views

Finding a matrix fixed by this permutation

I have an $n\times n$ matrix $M=1-\sigma(g)$ where $\sigma:S_n\to\mathbb{C}^n$ is a representation (reducible) of the symmetric group. I want to find another matrix $A(g)$ such that $MA(g)=0.$ ...
3
votes
1answer
103 views

Can we find subgroup of $(\mathbb{R},+)$ with order 2?

We used the following idea: first get a set of Hamel basis for $\mathbb{R}$, secondly, divide it into two parts such that one set of the Hamel basis forms a group, the other one is just the former one ...
3
votes
1answer
93 views

Irreducible subgroups of $\text{GL}(2,p)$

I just wonder any sources that provides the list of all minimal irreducible subgroups of $\text{GL}(2,p)$. Here the term "minimal" means there is no proper subgroup that is irreducible. A list of ...
7
votes
1answer
77 views

Finding the order of the automorphism group of the abelian group of order 8.

So I am given an abelian group of order $8$ such that for all non-identity elements $x^2 = e$ (all elements have order two). So I know the answer is gonna be $168$, but I gotta prove this. So far I ...
5
votes
3answers
134 views

Showing that if $fg=gf$ and $fh=hf$, then $gh=hg$, where $f$, $g$, and $h$ are affine functions

Given real numbers $a$ and $b$ ($a \ne 0$), let $f_{a,b}$ be the function $\mathbb{R} \to \mathbb{R}$ defined by $x \mapsto ax+b$. The set of such functions is a permutation group on $\mathbb{R}$, ...
11
votes
3answers
112 views

Order of matrices in $GL_2(\mathbb{Z})$

Let $A\in GL_2\left(\mathbb{Z}\right)$, the group of invertible matrices with integer coefficients, and denote by $\omega(A)$ the order of $A$. How we prove that $$\left\{\omega(A);A\in ...
5
votes
0answers
88 views

How is $\mathrm{PGL}(V)$ a subgroup of $\mathrm{P\Gamma L}(V)$?

I've stumbled upon a strange exercise while reading "Notes on Infinite Permutation Groups" by Bhattacharjee, Möller, Macpherson and Neumann. If you have the book, the exercise is 7(ix) on page 66. ...
3
votes
1answer
65 views

Are the irreps of SO(4) necessarily real?

I'm not a group theory buff, but I was confused when I used some code to generate the irreducible representatives of SO(4) and found the resulting matrix elements to be complex. Is this possible, or ...
1
vote
1answer
42 views

Finding $A,B\in SL_2(\Bbb{Z})$ of finite order with the property that $AB=C$ where the order of $C$ is infinite.

I'm trying to think of two matrices $A,B\in SL_2(\Bbb{Z})$ of finite order ($A^n=B^m=I$) with the property that $AB=C$ where the order of $C$ is infinite. I guess that just by trial and error I ...
1
vote
1answer
76 views

Determining which maps are isomorphisms

1) Let $G=G'=\{(a,b)\mid a,b \in \mathbb{R}, a, b \ne 0\}$ with group operation $(a_1,b_1)(a_2,b_2)=(a_1a_2,b_1b_2)$. Let $\phi (a,b) = (b^{-1}, ab^2)$. My solution: 1-1: Suppose ...
1
vote
2answers
113 views

Elements in $\text{GL}(n,q)$ with irreducible characteristic polynomial

Let $x,y\in\text{GL}(n,q)$ be of the same order such that both the characteristic polynomials of $x,y$ are irreducible. Must $x,y$ always be conjugate in $\text{GL}(n,q)$? More restrictly, I am ...
0
votes
2answers
47 views

Showing the Function in Second Isomorphism Theorem is Well-Defined

In particular this is done in the context of vector spaces (so I'm using abelian notation). Let $S$ and $T$ be subspaces of $V$. I am trying to show that $(S + T)/T \cong S/(S \cap T)$. Then we ...
2
votes
1answer
51 views

Does a linearly independent combination of $n$ group elements generate $\mathbb{Z}^n$ as a group?

I am working through some group theory stuff and I have a question for you guys. I know that $\mathbb{Z}^n$ is a vector space, and a group. Now if I think of it as a vector space, I know a linearly ...
2
votes
1answer
24 views

quotient representation — show that it actually is a linear representation

Let $U \subset V$ be a $G$-invariant subspace and $\rho : G \to \mathrm{GL}(V)$ a linear representation. Show that $\rho_{U} : G \to \mathrm{GL}(V/U)$ with $\rho_U(g)(v+U) = \rho(g)(v) + U$ is a ...
2
votes
2answers
34 views

dual representation — show that it actually is a linear representation in the dual space

Let $\rho : G \to \mathrm{GL}(V)$ be a linear representation in $V$. Show that $\rho^* : G \to \mathrm{GL}(V^*)$ with $\rho^*(g)(f) = f \circ \rho(g^{-1})$ is a linear representation on $V^*$. So ...
7
votes
1answer
111 views

Probability that a $3\times 3$ matrix with entries in $\{0,1,2,3\}$ is invertible.

Let $A$ be a $3\times 3$ matrix, and each of its entries takes value from $\{0, 1, 2, 3\}$ with probability $1/4$ for each value. What is the probability that A is invertible? I have tried to list ...
3
votes
2answers
173 views

Size of conjugacy classes in $GL(4,2)$

I'm asked to find out all of the conjugacy classes, their order and their size for $GL(4,2)$. Finding representatives is possible by looking for all the rational canonical forms over the field and ...
2
votes
1answer
38 views

Torsion free divisible group

I have some confusion about the torsion free divisible abelian group (but not the free abelian group).This group is a $\mathbb Q$ vector space. If $G$ has a finite dimension as a vector space then can ...
6
votes
2answers
100 views

The Center of $GL(n,k)$

The given question: Let $k$ be a field and $n \in \mathbb{N}$. Show that the centre of $GL(n, k)$ is $\lbrace\lambda I| λ ∈ k^∗\rbrace$. I have spent a while trying to prove this and have succeeded ...
2
votes
1answer
93 views

When does there exist a $g\in G$ such that $H = gH'g^{-1}$?

Suppose $H$ and $H'$ are two finite subgroups of $G$. Do there exist theorems that give conditions so that $H$ and $H'$ are conjugate, that is, there exists $g\in G$ such that $H = gH'g^{-1}$? Edit: ...
3
votes
1answer
63 views

Definition of $\text{GL}(n,R)$

How do one usually define the general linear group over a ring $R$, denoted by $\text{GL}(n,R)$. I was told in a paper that $\text{GL}(n,R)$ is a group, and I presumed that $$\text{GL}(n,R)=\{A\in ...
3
votes
5answers
123 views

Order of general linear group of $2 \times 2$ matrices over $\mathbb{Z}_3$

From problem 2.3.25 in Topics in Algebra, 2$\varepsilon$ by I. N. Herstein: Let $G$ be the group of all $2 \times 2$ matrices $\left(\begin{array}{c c}a & b \\ c & d\end{array}\right)$ ...
6
votes
2answers
130 views

Exponent of $GL(n,q)$.

Another exponent problem. $GL(n,q)$ is the group of invertible $n\times n$ matrices over the finite field $GF(q)$, where $q$ is a prime power. I am trying to figure out the exponent of this group. ...
2
votes
0answers
60 views

“unitary group” with respect to non-hermitian matrices?

the group $GU(n,q)$ is usually defined as the group of $n$ by $n$ matrices $X$ over $\mathbb{F}_{q^2}$ such that $X^H X=I_n$, where $I_n$ denotes the identity matrix and $^H$ the hermitian transpose ...
-3
votes
2answers
126 views

Verify that a set is a group under $2 \times 2$ matrix multiplication

I need to verify that the set $\left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right)$ where $a, b, c, d \in\mathbb{R}$, $ad - bc = 1$ is a group under $2 \times 2$ ...
5
votes
4answers
183 views

What is an additive group?

Is an additive group a group which only has an addition operation, or can it also have other operations on it? Thanks

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