3
votes
1answer
26 views

Schur's Lemma: Is the isormorphism between two irreducible spaces unique?

Suppose $V_1 \neq V_2$ are two irreducible representations of the finite group G. Then Schur's Lemma says that any G-invariant map between them is either 0 or an Isormorphism. I understand that if ...
2
votes
2answers
57 views

How to see that $PSU(2)$ is same as $SO(3)$?

Some background: We have an action of $SU(2)$ on the space of traceless Hermitian matrices, $\mathcal{H}$, via conjugation: $$SU(2)\times \mathcal{H}\to \mathcal{H}, \ (U,H)\mapsto UHU^{-1}.$$ The ...
0
votes
2answers
48 views

Permutation matrices for symmetry group $O_h = S_4 \times C_2$

Does anyone know of a quick way to enumerate the permutation matrices for the symmetry group of the cube $O_h = S_4 \times C_2$? $O_h$ has $48$ elements; if we label the vertices of the cube $1,2,…,8$ ...
9
votes
4answers
116 views

If permutation matrices are conjugate in $\operatorname{GL}(n,\mathbb{F})$ are the corresponding permutations conjugate in the symmetric group?

There is a standard embedding of the symmetric group $S_n$ into $\operatorname{GL}(n,\mathbb{F})$ (for any field $\mathbb{F}$) that sends each permutation in $S_n$ to the corresponding permutation ...
1
vote
1answer
40 views

symmetric group as a subgroup of general linear group

Is the symmetric group $S_n$ a normal subgroup of the general linear group $GL(n,\mathbb{R})$? We regard $\sigma\in S_n$ acts on $\mathbb{R}^n$ by permuting the coordinates ...
3
votes
1answer
34 views

Every representation of a finite group is reducible?

I somehow "proved" that every representation of a finite group is reducible. While I'm fairly sure the error is something silly, I can't seem to place it. Could someone please help me figure out what ...
4
votes
1answer
49 views

Same quadratic forms on $\mathbb R^n$

Let $q$ be an inner product on $\mathbb R^n$ and $Q$ be its matrix expressed in the canonical basis of $\mathbb R^n$. Assume that the group $$SO(q)=\{A\in M_n(\mathbb R) \ | \ A^TQA=Q\}$$ of ...
1
vote
0answers
33 views

Give a $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$

Consider $SL_{2}(\Bbb Z_p)$ if q & p be two primes, $p>q$. Give an example of a subgroup $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$ when i) $q|(p-1)$ ii) $q|(p+1)$
0
votes
0answers
31 views

Semi direct product

Prove that (i) $GL_n(R)= \coprod_{w\in S_n} UwB$ where $w \in S_n$ is a permutation matrix. and $U$ is a subgroup of $GL_n(R)$ consisting of upper triangular matrices with diagonal entries $1$ and ...
5
votes
0answers
47 views

Counting the number of elements in a double coset

Let $G$ denote the groups of $n\times n$ invertible matrices and $H$ be the subgroup of invertible upper triangular matrices. For $n=2$, by row reduction, or equivalently LU decomposition, it is ...
0
votes
0answers
23 views

Study the associative and commutative properties and neutral and inverse elements of these groups

Group m*n = max(m,n) on Z and N So i showed its associative by m,n,p in Z and (m*n)*p = max(m,n)p =max(m,n,p) And m(n*p) = m*max(n,p) = max(m,n,p) Commutative m*n = max(m,n) and n*m = max(n,m). I ...
2
votes
2answers
34 views

Is an elementry abelian group a non-degenerate symplectic vector space?

Let $A$ be an elementry abelian group with $|A|=p^{n}$ where $p$ is a prime number and $n$ is even. It is well-known that we can consider $A$ as a vector space of dimension $n$ over the field $F_p ...
1
vote
1answer
43 views

Endomorphism ring of finite-dimensional representation

If $G$ is any group and $V$ is a finite-dimensional representation of $G$, then we can form the endomorphism ring $E = End_G(V)$. Suppose that $V$ is indecomposable, i.e. not a direct sum of ...
0
votes
1answer
41 views

Prove that the set of all unitary operators on unitary space $ \mathcal U$ is a group

Prove that the set of all unitary operators on unitary space $ \mathcal U$ is a group, in particular sub-group in group of invertible linear transformations on $ \mathcal U$. Our course is ...
1
vote
1answer
86 views

Is $SL_2(\mathbf{Z}/n\mathbf{Z})$ generated by the elementary matrices?

By the elementary matrices I mean the matrices with diagonal elements $1$ and at most one nonzero element off-diagonal. I have seen it claimed that this is true, but no proof was given. I know that ...
1
vote
1answer
29 views

Group Theory $Z_2$ representations

I am trying to understand some group theory. In the notes I am following, I am told: Recall the representations of $\mathcal{Z}_2$: Trivial: $\rho_0(e) = 1$, $\rho_0(a)$ = 1 (i) $\rho_1(e) = 1$, ...
6
votes
4answers
163 views

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$?

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$? I wrote it down in an imprecise way on purpose. The notation above is the linear algebra one: ...
2
votes
3answers
58 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
100 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
128 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
64 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
101 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
29 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
44 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
75 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
115 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
70 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
61 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
42 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
110 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
46 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
45 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 ...
7
votes
1answer
92 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
61 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
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 ...
2
votes
1answer
63 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
45 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
161 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
18 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
36 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
83 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
51 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
49 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
55 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
23 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. ...