5
votes
1answer
56 views

If $\{M_1,M_2,M_3,\dots,M_r\}$ is a multiplicative group of matrices, and $\sum_{i=1}^r tr(M_i) =0$, then $\sum_{i=1}^r M_i =0$

Let $\{M_1,M_2,M_3,\dots,M_r \}$ be set of real $n\times n$ matrices which forms a group under matrix multiplication. If $\displaystyle \sum_{i=1}^r tr(M_i) =0$, prove that $\displaystyle ...
3
votes
2answers
125 views

Prove that there is a basis of a lattice $\Lambda$ s.t. a reflection is of a certain form

Consider $\Lambda$ a lattice in $\mathbb{R}^2$. Let $S \in O(\Lambda)$ be a reflection, i.e. $\det S = -1$. Set $S_{1}= \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}$ and $S_{2}= ...
2
votes
0answers
51 views

Show $\exists x\in\mathbb{R}^2$ such that $g(x)\neq x$ for all $1\neq g\in G$ for a movement [closed]

Let $G\subset B_2$ ($B_n$ is a movement $\beta:\mathbb{R}^n\to\mathbb{R}^n$ which preserves the norm $||v-w||=||\beta v-\beta w||$ and is also bijective) a discrete subgroup (The subgroup ...
2
votes
3answers
45 views

$SL(V)$ and $PSL(V)$ act $k$-transitively on the space of all $1$-dimensional subspaces.

A group $G$ acts $k$-transitive on some set $X$ if for every two $k$-tupels $(x_1, \ldots, x_k)$ and $(y_1, \ldots, y_k)$ there exists some $g \in G$ such that $$ g\cdot x_1 = y_1, \ldots, g\cdot ...
2
votes
1answer
44 views

Order of $ g= \big(\begin{smallmatrix} \ 1 & 1 \\ 1 & 0 \end{smallmatrix}\big)\in GL_2(\mathbb F_3)\;. $

Let\begin{align*} g= \begin{pmatrix} \ 1 & 1 \\ 1 & 0 \end{pmatrix}\in GL_2(\mathbb F_3)\;. \end{align*} Its minimal polynomial is $P_g(X)=X^2-X-1$ which divides $X^8-1$ in $\mathbb F_3[X]$, ...
4
votes
2answers
76 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} ...
2
votes
1answer
37 views

Non-obvious deduction regarding conjugates in $\text{GL}_2(\mathbb{F}_p)$

Let $\text{GL}_2(\mathbb{F}_p)$ act on $\mathbb{F}_p^2,$ the set of $2$-vectors with entries in $\mathbb{F}_p$, by matrix multiplication. $[\,$Prove that$\,]$ for any ...
3
votes
2answers
53 views

The number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$

How to count the number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$? I know that the number of invertible matrices is GL$(n,p)$. Have any ideas?
1
vote
1answer
86 views

elementary row operations

We know that the elementary row operations generate the general linear group. Suppose that we have a subset of elements of a given general linear group. Is it possible to generate given general linear ...
1
vote
1answer
37 views

Find position on number in sorted array

I am trying to calculate some thing and I got lost. I have sorted(low to high) array of $N$ numbers, with first number $K$ and numbers sum of $S$. Assuming that there are no duplicated numbers and ...
0
votes
0answers
46 views

Matrix coefficients of representations of finite groups

In finite-dimensional complex representations of finite groups, I would like to understand what I can learn by looking at a single matrix coefficient. In particular, I would like to look at "diagonal" ...
1
vote
1answer
102 views

Elements of order 3 in $PGL(4,\mathbb{R})$

I need to classify all elements of order 3 up to conjugation in $PGL(4,\mathbb{R})$. It's sufficient to give a representative of each conjugacy class. My thoughts: consider instead $GL(4,\mathbb{C})$ ...
5
votes
4answers
89 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
91 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 ...
2
votes
1answer
41 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 ...
0
votes
0answers
57 views

automorphism group

an endomorphism of a vector space V is a linear operator V → V. and an automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is ...
1
vote
1answer
46 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
56 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 ...
0
votes
1answer
41 views

decompose a real representation of a group $C_2\times C_2

Let $\Phi$ be a real representation of the group $C_2\times C_2=\{e,a\} \times \{e,b\}$ such as $ \Phi(a)=\begin{bmatrix}5 & -4 & 0\\6 & -5 & 0 \\0 & 0 & 1\end{bmatrix}$ and $ ...
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 ...
3
votes
2answers
276 views

Existence of a G-invariant matrix

Let $\phi: G \to GL(\mathbb{R}^n)$ be a homomorphism, $G$ finite. Prove that there is a positive-definite matrix $M$ such that $\phi(g)^tM \phi(g) =M$ $\forall g \in G $. This looks really ...
3
votes
0answers
84 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 ...
0
votes
0answers
33 views

The group generated by all permutations of an orthonormal basis and their negatives

Consider the group generated by all permutations of an orthonormal basis in $\mathbb{R}^n$, and by taking the negatives of these basis vectors. The general element of this group is a matrix (in the ...
1
vote
1answer
44 views

Group algebras, Maschke's lemma and direct sums of matrix algebras

Let $G=\{g_1,g_2,\dots,g_n\}$ be an arbitrary finite group. We consider its representations over $\mathbb{C}$. There is Maschke's theorem which states that each representation of $G$ is a direct sum ...
4
votes
1answer
110 views

Conjugacy classes and orders of matrices.

The following are prime decompositions in $\Bbb{Z}_7[x]$: $x^8+1= (x^2-x-1)(x^2+x-1)(x^2+3x-1)(x^2+4x-1)$ $x^4+1= (x^2+3x+1)(x^2+4x+1)$ (a) Give representatives for the conjugacy classes of ...
2
votes
4answers
145 views

Irreducible representation of dimension $5$ of $S_5$

i am searching for a concrete as possible description of the (there are two but the are obtained from each other by tensoring with the signature representation) irreducible representation of dimension ...
6
votes
0answers
258 views

Finding the maximum number of subspaces of a vector space over finite field that satisfy these relations

I have a question and I am stuck. I was wondering if anyone has a thought, before I start a brute-force search. For $q$ a prime number and $n =6$, let $\mathbb {F}_{q}^{n}$ be an $n$-dimensional ...
11
votes
1answer
137 views

Abelian subgroups of $GL_n(\mathbb{F}_p)$

Let $p$ be a prime number, and let $k=\mathbb{F}_p$ be the field of $p$ elements. Let $G=GL_n(k)$. We know that $$|G|=\prod_{i=0}^{n-1}(p^n-p^i)=p^{\binom{n}{2}}\prod_{i=0}^{n-1}(p^{n-i}-1)$$ so ...
6
votes
2answers
79 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
3answers
201 views

Sylow $p$-subgroups of Finite Matrix Groups

Let $G$ be a subgroup of $GL_n(\mathbb{F}_p)$, with $n\le p$, and let $P$ be a Sylow $p$-subgroup of $G$. Do all non-trivial elements of $P$ have order $p$? I believe the answer is yes, because I ...
4
votes
1answer
211 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 ...
8
votes
1answer
174 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 ...
2
votes
1answer
102 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: ...
2
votes
2answers
124 views

Given $v, w$ find a matrix $P$ such that $v = Pw$

How can I show that given two non-zero vectors $v, w \in \mathbb{F}_q^2$ there exists a matrix $P \in SL_2(\mathbb{F}_q)$ such that $v = Pw$?
7
votes
2answers
251 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
70 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 ...
5
votes
1answer
94 views

Is it possible to reverse this sequence of permutations?

Let $ S = (a_1, a_2, ..., a_N) $ be a finite (arbitrarily long) sequence of elements, and let $p_1, p_2, ..., p_n $ be the first $n$ prime numbers, with $n \ge 3$. We apply a sequence of ...
2
votes
1answer
36 views

How can I show that $ASL_n(F)$ is acting 2-transitively?

One of my friends asked me to ask this question here. This is a question from his last exam: Let $$ASL_n(F)=\{T_{A,v}:V_n(F)\to V_n(F)\mid\exists A\in SL_n(F), \exists v\in V_n(F), ...
2
votes
1answer
209 views

Proof that cofactor expansion has unique value

Edit: I'm genuinely not sure why this has gotten little activity. If somebody knows, please tell me, so I can rework it. As a note: I am a purist, and really want to see a proof of this, but I've ...
1
vote
1answer
67 views

Isomorphisms of linear representations of finite groups

Let $G$ be a finite group with representations $\rho_1, \rho_2:G\rightarrow GL(V)$. According to the definition of representation isomorphisms, $\rho_1$ and $\rho_2$ are isomorphic if there exists a ...
0
votes
1answer
57 views

Commutator group of parabolic subgroups of $GL_n(q)$

Let $q$ be a prime power, $G=GL_n(q)$ and $P=q^{km}{:}(GL_k(q) \times GL_m(q))$ be a parabolic subgroup of $G$, where $k+m=n$. What is the commutator group $P'$ of $P$?
3
votes
1answer
153 views

Linear algebra of finite abelian groups

Let $\phi:G \to H$ be a surjective homomorphism of finite abelian groups, and let $g_1, \ldots, g_n$ be an irredundant set of generators (from now on, a basis) for $G$. be a basis for $G$, meaning a ...
1
vote
1answer
137 views

Dimension of subspace fixed by subgroup representation.

If $G$ is an abelian group with cyclic subgroup $H$ and $(\rho,V)$ is a (permutation) representation of $G$. Then I can form a representation of $H$ by considering the composition ...
4
votes
3answers
206 views

Symmetric group [duplicate]

Possible Duplicate: What kind of “symmetry” is the symmetric group about? Could you tell me please, why Symmetric group is called "symmetric"? I found an example with quadrate, ...
1
vote
0answers
79 views

Sets of independent vectors

So we're working in Z2k, the group of bit-vectors of length k and componentwise addition modulo 2. Now I'm trying to make a function yj=1..?(vi) assigning elements of Z2k to n vertices, such that ...
3
votes
2answers
163 views

Proper subgroups of $PSL(3,3)$ are solvable

There is an exercise in a linear group's book that: prove that every proper subgroup of $PSL(3,3)$ is solvable. How can I prove that? the best.
10
votes
1answer
711 views

On the order of elements of $GL(2,q)$?

There's a particular property of the elements of $GL(2,q)$, the general linear group of $n\times n$ matrices over a finite field of order $q$, that I don't understand. First, I know that the order of ...
0
votes
0answers
143 views

Which dimensions can have groups as basis?

For which n is there a set of $n^2$ LI. $n\times n$ matrices, which form a finite group? I know $n=4$ is possible by forming products of Dirac's gamma matrices. These matrices are the identity plus ...
2
votes
1answer
81 views

Non conjugate $p$-subgroups of $\mathrm{GL}_n(\mathbb Z)$

It is probably not that difficult but I can't find an example of two non-conjugate $p$-subgroups (of same order) of $\mathrm{GL}_n(\mathbb Z)$ ($n>1$).
1
vote
2answers
506 views

What are the finite subgroups of $SU_2(C)$?

Is there any reference which classifies the finite subgroups of $SU_2(C)$ up to conjugacy ? What I know is that lifting the finite subgroups of $SO_3(R)$ by the map $SU_2(C) \rightarrow PSU_2(C)$ ...