1
vote
0answers
14 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 ...
1
vote
2answers
41 views

Can a the field of fractions or quotient field F of an integral domain R be free over some set as an R-module?

We know any integral domain R when extended to a quotient field F, then F is free as an F-module on the set {1}. Can this field be free over some set as an R-module.
2
votes
1answer
36 views

Matrix linear algebra generators

Linear algebra and special-linear group experts please help: It is known that in principle one can generate this $C$ matrix form the $A$ and $B$ matrix below. Here $$ C=\begin{pmatrix} 0& -1& ...
0
votes
0answers
40 views

double coset represntatives

Let $H$ be a subgroup of finite index in the group $G$. Let $g\in G$. We use $r\in HgH/H$ as notation for $g\in R$, where $R$ is a complete set of representatives for $HgH/H.$ Proof or disproof: If ...
3
votes
1answer
68 views

Generators of Special Linear Groups

Linear algebra and special-linear group experts please help: I learn that in principle one can generate this $M$ matrix form the $B_1$ and $B_2$ matrix below. Here $$ M=\begin{pmatrix} 0& 1& ...
1
vote
0answers
33 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
19 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
50 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
48 views

functions determined by charachters are linearly independent?

Let $X$ be a set with an action of $\mathbb{Z}/N\mathbb{Z}$. For a Dirichlet character $\chi$ mod N we set $$R(\chi)=\left\{ f:X \to \mathbb{C} ~\mid~ f(l s)=\chi(l)f(s) \text{ for all } l\in ...
4
votes
1answer
58 views

Is the determinant the “only” group homomorphism from $\mathrm{GL}_n(\mathbb R)$ to $\mathbb R^\times$?

This might be a dumb question; I know only enough group theory to be able to ask dumb questions. Ken W. Smith has pointed out that one way to get intuition about the determinant is to observe that it ...
2
votes
1answer
26 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
1answer
20 views

MATLAB: How to make a list of words in matrices?

Cross-posted from stackoverflow, where there seemed to be no interest: http://stackoverflow.com/questions/22329233/matlab-how-make-a-list-of-words-in-matrices Given two n by n invertible matrices A ...
2
votes
5answers
175 views

how to prove $e^{A \oplus B} = $$e^A$ $\otimes$ $e^B$ where A and B are matrices? (Kronecker operations)

how to prove $e^{A+B} = $$e^A$$e^B$ where A and B are matrices? The operations '+' and '*' are defined such that AI + IB = A+B, where I is the identity matrix. I suppose these are called Matrix ...
0
votes
0answers
20 views

Perturbations of Lattices

This is a little idea I've been playing with. I was wondering if anyone has seen anything similar or if they have any questions/insights: Let $\lambda:=\{\displaystyle\sum_{i=1}^{n}\alpha_i ...
2
votes
2answers
52 views

Dimension of $SO_n(\mathbb{R})$

Is there a simple proof that the dimension of $SO_n(\mathbb{R})$, a.k.a the group of rotations in $n$-dimensional space is $(n-1)n/2$? It would be great to see some proofs based only on the ...
1
vote
2answers
41 views

Is the group of all determinants of all invertible $n \times n$-matrices isomorphic to $\langle\mathbb{R}^*, \cdot\rangle$?

I am doing Linear Algebra and Abstract Algebra simultaneously, and in Linear Algebra class, going through determinants, I thought of something interesting (for a freshman just learning the subjects, ...
4
votes
4answers
77 views

Show that $Q_8$ can't be embedded in $M_{2 \times 2}(\mathbb{R})$ as a group.

So, suppose that we're working in a field $F$. Consider the ring $M_{n \times n} (F)$ which is the set all $n \times n$ matrices with entries in $F$. Is it possible to determine whether a matrix ...
2
votes
1answer
75 views

Does $GL(n,K)$ act transitively on $1$-dim subspaces of $K$

If we let $K$ be a field and $GL(n,K)$ act by right multiplication on the $1$-dim subspaces of $K^n$. Then if we take $\langle v_1 \rangle, \ldots \langle v_n \rangle \in K^n$ distinct and $\langle ...
0
votes
2answers
38 views

Properties of derived group

If K is a normal subgroup of G then does the following equality hold: $[G/K,G/K]=[G,G]/(K \cap[G,G]) $ If this is true then prove it and if not then give a counter example.
2
votes
0answers
34 views

projective geometry and projective space

Let $V$ is a vectorspace over field $F_q$, we denote the set of all subspaces of $V$ by $\mathcal{P}(V)$. I saw some referencess they mentioned $\mathcal{P}(V)$ as a projective space and some ...
2
votes
1answer
33 views

Embedding symmetric groups into general linear groups

Note that any finite abelian group $\Gamma$ can be embedded into $\operatorname{GL}(1,A)$ for some commutative ring $A$. Indeed, let $A=\mathbf Z[\Gamma]$ and $\Gamma\hookrightarrow ...
2
votes
1answer
93 views

Prove G is a group under matrix multiplication.

G is the set of matrices of the form $G=$$\begin{pmatrix} x & x \\ x & x \end{pmatrix}$. So for this set to be a group I know it needs to be: Closed under matrix multiplication The ...
0
votes
1answer
24 views

Quitient of two infinite-dimentional groups

Is it possible that quotient group G/H will be finite-dimensional while G and H are infinite-dimensional? I believe that the answer is YES, but couldn't find any example. Appreciate any hint.
0
votes
0answers
31 views

Infinite dimension spaces

Consider, doubly infinite row vectors $(a)=(\cdots, a_{-1}, a_0, a_1,\cdots)$ with $a_i's$ real form a vector space. Is this space isomorphic to $\mathbb{R^\infty}$. Here ...
0
votes
0answers
62 views

Why is the special linear group generated by elementary matrices that add a multiple of row$ j$ to row $i$

The general linear group is generated by elementary matrices that add a multiple of row $j$ to row $i$ and elementary matrices that multiply row $i$ by a scalar. This is because you can write an ...
1
vote
2answers
29 views

Does conjugation preserve spectrum of matrices?

Actually, I saw normalizer of diagonal matrices are permutation matrices. I read the answer but I don't know how to prove that conjugation preserves the spectrum. Actually I do some proof on 2x2 ...
1
vote
1answer
41 views

Unique Complete Reducibility of Finite Groups

Maschke's Theorem states that every complex representation $(\rho,V)$ of a finite group $G$ can be written as a direct sum of irreducible representations that form subsets of V, such that $V = ...
1
vote
1answer
46 views

Consequences of Schur's Lemma

Schur's Lemma states that given two irreducible representations $(\rho,V)$ and $(\pi,W)$ of a finite group $G$ (V and W are linear vector spaces over the same field), and a homomorphism $\phi\colon ...
1
vote
3answers
93 views

Determine whether or not the set of real numbers $\mathbb R$, together with the operation* defined by $a ∗ b = (a+b)/3$, forms a group. [closed]

How would one show this is associative? I think I've shown it is closed but I'm not sure how to show it is associative.
-1
votes
2answers
68 views

How is this a finite group? [closed]

Let $G$ be the set of all $2\times2$ matrices $\begin{pmatrix} a & b\\ c & d\\ ...
2
votes
0answers
44 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 ...
2
votes
3answers
83 views

Matrix-free proof of $Z(GL_n(F)) = \{\lambda I:\lambda \in F^\times\}$?

How does one prove that $$Z(GL_n(F)) = \{\lambda I:\lambda \in F^\times\}$$ without resorting to matrices (and bases)? (BTW, $Z(GL_n(F))$ is the center of $GL_n(F)$, the general linear group of order ...
2
votes
3answers
161 views

Rank-Nullity Theorem Proof.

Let $V$ and $W$ be linear vector spaces. Let $\theta$ be a linear map from $V$ to $W$. Why is $\dim(V) = \dim(\operatorname{Im}(\theta)) + \dim(\ker(\theta))$? I know that there is an isomorphism ...
-3
votes
2answers
127 views

Does the set of all 2x2 matrices with real entriesand determinant 2 form a group? [closed]

Does anybody know the answer to this question under the operation of multiplication? Also a brief explanation would be appreciated! Thanks.
1
vote
1answer
52 views

complex irreps is in bijective correspondence with sequences

Let $\{a_n\}$ be a sequences of positive integers such that $$0 \leq a_n\leq p^n - 1,$$ $$a_n \equiv a_{n +1} \bmod p^n \quad \text{for all $n$}$$ Prove that the complex irreps of the group $ ...
3
votes
3answers
50 views

irrep of a non unit element in the finite group

Let $G$ be a finite group. Prove following statement. $\forall g \in G$ such that $g \neq e$ $ \exists$ a complex irrep $\rho$ such that $\rho(g) \neq E$ I have no idea how to start it. I can prove ...
21
votes
1answer
253 views

Is a field determined by its family of general linear groups?

Assume that $K,L$ are fields such that there is an isomorphism of groups $\mathrm{GL}_n(K) \cong \mathrm{GL}_n(L)$ for all $n \in \mathbb{N}$. Does it follow that $K \cong L$? I am also interested in ...
1
vote
1answer
59 views

Schur's Lemma and division algebras

Let $A$ be an abelian subgroup of the unimodular group of degree $n$ (i.e. $GL(n,\mathbb Z)$). $A$ can be regarded as a group of automorphisms of a free abelian group of rank $n$ ($\mathbb Z^n$), and ...
2
votes
0answers
67 views

Soluble Linear Groups

Let $G$ be a group with a normal (invariant) series $1=G_n\leq ... \leq G_0=G$ whose factors are either torsion-free abelian group of finite rank, finite elementary abelian $p$-group or direct ...
1
vote
1answer
74 views

Constructing a rotation matrix from complex eigenvalues

I am trying to construct a rotation matrix $\mathbf{R}\in\mathbb{R}^{3\times3}$ rotating around an axis $\hat{n}$ in a basis $\{\hat{n},\hat{u}_{1},\hat{u}_{2}\}$. Formally: Given a basis ...
1
vote
0answers
39 views

“Fixed Domains” of a Linear Transformation

Given a linear transformation $T$, I need to find the set of all domains $D$ such that $T:D\mapsto D$. Equivalently, I need to find the set of all domains $D$ that are symmetric under $T$. Aside from ...
2
votes
0answers
25 views

Linear group action over an hermitian space.

Let $\mathrm{h}$ be an non degenerate hermitian form on $\mathbb{C}^2$, and suppose $\mathrm{h}$ is of signature $(1,1)$. Let $A$ and $ B \in \mathrm{SL}(2,\mathbb{C})$ such that $G$ the group ...
10
votes
2answers
133 views

Is $SO_n({\mathbb R})$ a divisible group?

The title says it all ... Formally, if $SO_n(\mathbb R)=\lbrace A\in M_n({\mathbb R}) |AA^{T}=I_n, {\sf det}(A)=1 \rbrace$ and $W\in SO_n(\mathbb R)$, is it true that for every integer $p$, there is ...
3
votes
1answer
95 views

Representation theory and direct sum

I came across the following theorem in one of the online notes regarding representation theory which I thought should have a simple proof. I am trying to prove it using basic linear algebra tools: ...
5
votes
1answer
66 views

Generator of End(V)

If V is a finite-dimensional vector space of dimension n and G⊂End(V) such that G generates End(V) meaning that any element of End(V) is expressible as a linear combination of products of a number of ...
0
votes
1answer
62 views

Equivalence relation $g\sim h :\Longleftrightarrow h \in \{g,g^{-1}\}$

Let $(G, \cdot)$ be a group with an Identity element $e$. (i) A relation on $G$ is defined through $g\sim h :\Longleftrightarrow h \in \{g,g^{-1}\}$. Show that $\sim$ is a equivalence relation and ...
3
votes
1answer
71 views

Linear transformation whose $n$th power is identity

Let $V$ be a vector space over field $F$ with $\dim_FV=2$. Suppose $T:V\longrightarrow V$ is a linear transformation with $T^n=Id$ for some positive integer $n$ (the finite $n$ is the order of $T$). ...
2
votes
1answer
29 views

Explicit formula for a right splitting once we have a left splitting

Assume we have a short exact sequence (of abelian groups or vector-spaces, it doesn't matter) $$0\rightarrow A\stackrel{\iota}\rightarrow B\stackrel{\pi}\rightarrow C\rightarrow 0.$$ If we have a ...
1
vote
2answers
82 views

Independent components of trace?

A $3 \times 3$ symmetric matrix has $6$ independent components: \begin{equation} \{ S_{ij} \} = \begin{pmatrix} S_{11} & S_{12} & S_{13} \\ S_{12} & S_{22} & S_{23} \\ S_{13} & ...
3
votes
0answers
78 views

Reflections in Dihedral Group

In Dihedral Groups, what is the meaning of reflection ? A line needs to be specified for a reflection to take place, but, if you specify only one line how will $D_n$ give all the symmetries for a ...