2
votes
3answers
49 views

When is a non-trivial homomorphism injective?

I noticed that over the natural numers $(\mathbb{Z},+)$ any group homomorphism $f : \mathbb{Z} \rightarrow \mathbb{Z}$ that is not the trivial one, is automatically injective. Where exactly does ...
1
vote
2answers
82 views

Show that $Z\times Z$ is not cyclic… [duplicate]

The full problem is as stated in the title. I am here to check if this is a valid proof. I thought it would be easiest using Linear Algebra. Recall that an infinite cyclic group is isomorphic to ...
2
votes
2answers
115 views

Set of sequences -roots of unity

Consider $G_n$ as the multiplicative cyclic group given by the $n^{th}$ roots of unity. $$G_n = \left\{ e^{ 2ik\pi/n} \mid 1\leq k \leq n \right\}$$ Now construct a sequence from each $G_n$ by ...
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 ...
6
votes
3answers
100 views

Automorphisms of $\mathbb{Z}/p\oplus\cdots\oplus \mathbb{Z}/p$

Consider the abelian group $$G = \underbrace{\mathbb{Z}/p\oplus\cdots\oplus \mathbb{Z}/p}_{n},$$ where $p$ is prime and $1\le n \le p$. I want to show that $G$ has no automorphism of order $p^2$. I ...
0
votes
0answers
25 views

Existence of Generalized Hadamard matrices

Does there exist a Generalized Hadamard matrix of order 20 over an Elementary abelian group of order 4 ,GH(20,EA(4))?
1
vote
1answer
48 views

Let $U_g$ be the group of units of $\mathbb{Z}/g\mathbb{Z}$ . . .

Let $U_g$ be the group of units of $\mathbb{Z}/g\mathbb{Z}$. Then $U_g$ is a subgroup of itself. For every unit $c$ of $U_g$, show the coset, $cU_g = U_g$. Show that the product of the elements of ...
1
vote
0answers
51 views

Does this subgroup of $\mathrm{SL}(2,\mathbb{C})$ have a a name?

The set of matrices $g$ characterized by $g=\begin{pmatrix}a&ib\\ ic&d\end{pmatrix}$, where $a,b,c,d \in \mathbb{R}$ and $ad+bc=1$, can be easily shown to be a subgroup of ...
3
votes
1answer
41 views

Computing the order of the first cohomology group $|H^1(S_n, \mathbb F_p^n)|$

Assume $n\geq 3$, $p$ is a prime, and that $S_n$ acts on $V=\mathbb F_p^n$ by permuting the basis vectors $v_1,\ldots, v_n$. I want to compute the order of the first cohomology group of this action. ...
0
votes
1answer
71 views

a potential application of the ping-pong lemma?

From my understanding, a simple result of the ping-pong lemma would state that if we have a set of linear transformations (matrices) $A_1,\ldots,A_n$ all of the same dimension, then if ...
9
votes
2answers
104 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 ...
0
votes
2answers
45 views

How to prove determinant is a group homomorphism and onto?.

I posted this question I am struggling with previously but it was put on hold for lack of context, I hope this is now clearer. Consider the determinant function Det: Mn($\mathcal{F}$) $\to$ ...
1
vote
1answer
48 views

How to prove the determinant is a group homomorphism

Consider the determinant function Det:M$_{n}(\mathcal{F})$$\rightarrow\mathcal{F}$, where $\mathcal{F}$ is a field. i) Explain how to restrict the domain and range of Det to obtain a group ...
0
votes
1answer
41 views

How to prove that two groups are isomorphic?

I am unsure how to do this question please help. Let $G$ be the subset of M$_{3}(\mathbb{R})$ defined by $G = \left\{ \left(\begin{array}{ccc} 1 & t & t^{2}/2\\ 0 & 1 & t\\ 0 & ...
0
votes
1answer
72 views

Abstract algebra true or false answer check

Sorry about the giant picture file, but typing up this many questions on Latex would take forever. My attempts are below, I am fairly sure 16+ are right My answers: -1T- -2T- -3F- -4F- -5T- -6F- ...
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 ...
3
votes
2answers
40 views

Show $D_3\cong S_3$ and $D_n\ncong S_n$ for $n\gt 3$

Show that $D_3\cong S_3$ and $D_3\ncong S_3$ for $n\gt 3$, where $D_3$ denotes the dihedral group and $S_3$ the symmetric group. I define a group isomorphism between $D_3$ and $S_3$. Both group ...
6
votes
1answer
79 views

Finite Subgroups of $GL_2(\mathbb Q)$

I want to prove that the only finite subgroups of $GL_2(\mathbb Q)$ are $C_1, C_2, C_3, C_4, C_6, V_4, D_6, D_8,$ and $D_{12}$. First, we determine all possible finite orders of elements. Now, an ...
1
vote
2answers
60 views

Having trouble understanding what ker f is

I am trying to understand what exactly 'ker f' is. My guess is that it is a set of all of the elements that were lost in the process of a mapping. For example: $f:A\to B$ $f = \left\{ \begin{align} ...
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
58 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 ...
2
votes
1answer
62 views

under what conditions a product of matrices is the identity matrix (more complicated than that)?

I have a set of matrices $A_1,\ldots,A_n$ and another set $B_i = A_i^{-1}$ for $i = 1,\ldots, n$ (I assume $A_i$ are invertible). Let $\mathcal{A} = \{A_i\} \cup \{B_i\}$. What are some simple ...
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 $\mathbb F_3[X]$, ...
2
votes
3answers
157 views

When to use $\times$ and $\otimes$

Im wondering when to use $\times$ and when to use $\otimes$. In some cases it seems very straightforward, for example $\times$ can be used when combining two elements into an n-tupel (for a product ...
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 ...
2
votes
1answer
16 views

How to show that the index of an abelian subgroup's annihilator is equal to the order of the subgroup?

Given a finite abelian group $G$, the dual group, $\hat {G}$, is the group of homomorphisms from $G$ into the multiplicative group of the roots of unity in $\mathbb C$. Now, by the Fundamental ...
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 ...
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
48 views

Computing quotients of abelian groups

Suppose that $A \cong \oplus_{i = 1}^{n} Z_{p_{i}^{k_{i}}}$ is some finite abelian group, and $(a_1, a_2, \ldots a_n)$ generates a subgroup $N$. If $\langle (a_1, a_2, \ldots a_n) \rangle$ was a ...
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 ...
1
vote
1answer
41 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 ...
2
votes
2answers
51 views

$SO(\mathbb{Q})$ is not a finitely generated group for $n \geq 2$

Suppose that $SO(\mathbb{Q}) = \{ A \in M_n{\mathbb{(Q)}}: A^TA=I, \det A = 1 \}$ is a subgroup of $n\times n$ matrices with rational entries under matrix multiplication. Show that for $n \geq 2$ ...
1
vote
1answer
21 views

Existence of neutral element at a certain position in subgroups

Given a group $G$ with neutral element $e$ and a subgroup $H \leq G$ as well as the equivalence relation $g_1 \sim g_2 \Leftrightarrow g_2^{-1} g_1 \in H$ (equivalence classes $[g]$). G be finite. ...
0
votes
1answer
42 views

How to express $x_{ij}$ using $x_{k,k+1}$?

Let $$ x_{i,j}(t) = \left( \begin{matrix} 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & \cdots & 0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 ...
1
vote
1answer
55 views

Finding homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{6}$.

Find all homomorphisms from $\mathbb Z_{12}$, the cyclic group of order $12$, to $\mathbb Z_6$. For each homomorphism $f\colon \mathbb Z_{12}\to \mathbb Z_6$, determine the kernel $\ker(f)$ and the ...
1
vote
2answers
32 views

Shown that this matrix is a representative of a group.

I have to show that this matrix (for which $ad - bc = 1$) \begin{pmatrix} a & b & 0\\ c & d & 0 \\ 0&0&1 \end{pmatrix} is a representative of a group. The same for this one: ...
4
votes
1answer
47 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
61 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
42 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
45 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
114 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
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
21 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
56 views

Functions determined by characters are linearly independent?

Let $X$ be a set with an action on $\mathbb{Z}/N\mathbb{Z}$. For a Dirichlet character $\chi \pmod N$ 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
2answers
127 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
30 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
21 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
187 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
23 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 ...