All Questions
Tagged with group-isomorphism matrices
44 questions
-1
votes
2
answers
110
views
Is $\mathbb{R}^4$ isomorphic to $\mathbb{C}^2$, $\mathbb{R}^4$ isomorphic to $\mathrm{Mat}_{4 \times 4}(\mathbb{R})$ as $\mathbb{R}$-vector spaces?
Is $\mathbb{R}^4$ isomorphic to $\mathbb{C}^2$ as $\mathbb{R}$-vector spaces?
Is $\mathbb{R}^4$ isomorphic to $\mathrm{Mat}_{4 \times 4}(\mathbb{R})$ as $\mathbb{R}$-vector spaces?
For the first one, ...
- 1
1
vote
0
answers
59
views
Is there a way to describe the map from $\mathbb Z^{n}$ to ${\rm Span}_{\mathbb Z} \{ Q(\mathbb Z^{n}), Q^{-1}(\mathbb Z^{n})\} $ using Q?
Suppose we have a non-singular rational matrix $Q$, consider the the $\mathbb{Z}$-span of the columns of $Q$ and $Q^{-1}$, denote it as $H = {\rm Span}_{\mathbb Z} \{ Q(\mathbb Z^{n}), Q^{-1}(\...
- 1,723
5
votes
1
answer
221
views
Is $PGL(2,\mathbb R)$ isomorphic to $SO(1,2)$?
Consider the following representation $\rho:GL(2,\mathbb R)\to GL(3,\mathbb R):G\mapsto \hat{G}$ where
$G=\begin{pmatrix}
\alpha&\beta\\
\gamma&\delta
\end{pmatrix}\in GL(2,\mathbb R)$
and
$\...
0
votes
1
answer
62
views
Are two matrices isomorphic? (as rings and as group) [closed]
Assume that $M_2(R) , M_3(R)$ are matrices with real cells $2 \times 2$ , $3 \times 3$ respectively.
1)Are $M_2(R) , M_3(R)$ Isomorphic as rings under addition and multiplication ? why?
2) Are $M_2(R) ...
- 3
1
vote
1
answer
44
views
Find $d$ such that $(M_d, \cdot)$ group is isomorphic to the $(\mathbb{C}^*, \cdot)$?
Let $d$ be an arbitrary real number and
$M_d=\left\{
\begin{pmatrix}
a& db\\
b& a
\end{pmatrix}\in\mathcal{M}_2(\mathbb{R}), \text{where } a^2-db^2\neq 0\right\}.
$
The problem is: show that $...
2
votes
1
answer
198
views
$\operatorname{GL}_n(k)/k^\times \cong\operatorname{SL}_n(k)$
In a lot of references, people mention that $\operatorname{SL}_n(k)$ is a normal subgroup of $\operatorname{GL}_n(k)$ and $\operatorname{GL}_n(k)/\operatorname{SL}_n(k) \cong k^\times$ via the ...
- 2,627
1
vote
4
answers
132
views
No determinant of a homomorphism even when $\dim V = \dim W$?
Why is there no determinant of a homomorphism $f: V \to W$, even when $\dim V = \dim W$?
I don't know how the concepts in linear algebra are related to each other and I am new to it.
Thanks
- 87
1
vote
1
answer
27
views
For any field $F$, prove that $\Sigma(2,F)$ is isomorphic to ${\rm Aff}(1,F)$, where $\Sigma(2,F)$ denotes the stochastic group.
$$\Sigma(2,F)=\left\{\begin{bmatrix}a&b\\1-a&1-b\end{bmatrix},a,b\in F\right\}$$
$${\rm Aff}(1,F)=\left\{\begin{bmatrix}a&b\\0&1\end{bmatrix},a\neq 0,a,b\in F\right\}$$
First I tried ...
- 366
2
votes
0
answers
56
views
Model-theoretic proof for $\mathrm{GL}_{n}(K) \simeq \mathrm{GL}_{m}(K) \Rightarrow n = m$
The statement in the title is true for any field $K$ (as far as I know), and the proof wasn't that hard (as far as I know again). After I read about the sketch of the proof of Ax-Grothendieck theorem ...
- 15.5k
5
votes
0
answers
142
views
Deciding if $\mathbb{Z}\ltimes_A \mathbb{Z}^5$ and $\mathbb{Z}\ltimes_B \mathbb{Z}^5$ are isomorphic or not
I have the two following groups
$G_A=\mathbb{Z}\ltimes_A \mathbb{Z}^5$, where $A=\begin{pmatrix} 1&0&0&1&0\\0&-1&0&0&0\\0&0&-1&0&0\\0&0&0&0&...
- 1,946
0
votes
0
answers
51
views
the isomorphism between two matrix groups [duplicate]
is there any isomorphism between two $n{\times} n$ matrix groups with inverse? If yes, what the isomorphism function?.
I have read that the $n{\times}n$ group matrix or $GL(n,\Bbb C)$ is isomorphic ...
4
votes
1
answer
127
views
Subgroups of $\text{SL}_2(p)$ isomorphic to $C_3 \rtimes C_4$
Let $p$ be a prime number. I am interested in classifying all subgroups $H \subset \text{SL}_2(p)$ such that $H \cong C_3 \rtimes C_4$, where $C_3$ and $C_4$ denote the cyclic groups of order 3 and 4 ...
- 2,074
-1
votes
1
answer
778
views
Proving the Heisenberg Group of modulo $p$ is isomorphic to $D_8$. [closed]
Prove the Heisenberg Group of modulo $p$ is isomorphic to $D_8$.
I'm having trouble specifically figuring out the way $D_8$ can be related to the unipotent upper triangle matrices with entries in $\...
- 101
1
vote
2
answers
140
views
Show that $\operatorname{Aff}(3)$ is isomorphic to $S_3$, the symmetric group of all permutations of 3 objects.
Show that $\operatorname{Aff}(3)$ is isomorphic to $S_3$, the symmetric group of all permutations of 3 objects.
Where
$$\operatorname{Aff}(3):={\{( \begin{array}{cc}
a & b \\
0 & 1
\end{...
- 852
0
votes
1
answer
2k
views
Isomorphism between $GL_2(\mathbb{F}_2)$ and $S_3$ [duplicate]
The question is to show that $GL_2(\mathbb{F}_2)$ and $S_3$ are isomorphic where $S_3$ is the symmetric group of $\{1, 2, 3\}$ and the group operation is composition.
I have listed all the elements ...
- 145
0
votes
0
answers
236
views
prove that the group$ R^n $ is isomorphic to the group $(Diag(n, R),+)$
Show that, the group $R^n $is isomorphic to the group$ (Diag(n, R),+)$ consisting of $n\times n$ diagonal matrices having positive real diagonal entries.
Recall: the group operation on $R^n$ is the ...
0
votes
2
answers
107
views
Prove $G/H \cong (\mathbb{R}-\{0\})\times(\mathbb{R}-\{0\})$
Let $G$ be the group of all real matrices of the form $\displaystyle\left( \begin{smallmatrix} a & b \\ 0 & c \end{smallmatrix} \right)$ with $ac \neq 0$ under matrix multiplication. Let $H$ ...
- 555
1
vote
0
answers
32
views
Is $H/L$ isomorphic to $K^*\times K^*$ where $K^*=(K_{\ne0},\cdot)$ with $K$ field and $H, L$ certain subgroups of $GL(2,K)$?
Let $K$ be a field, $H=\left\{\begin{bmatrix}a&b\\0&d\end{bmatrix}:a,b,d\in K, ad\ne0\right\}$, $L=\left\{\begin{bmatrix}1&b\\0&1\end{bmatrix}:b\in K\right\}.$
I'm asked to prove that ...
- 472
5
votes
2
answers
892
views
If S is a normal subgroup, identify the quotient group G/S. What are the $\varphi(G)$'s?
The following is an exercise from Artin's Algebra: (Kiefer Sutherland's voice)
Let G be the group of upper triangular real matrices $\begin{bmatrix}
a & b\\
0 & d
\end{bmatrix}$ with a and ...
user198044
9
votes
3
answers
743
views
Affine Transformations isomorphic to Heisenberg group
I want to show that the lie group $G$ of affine transformations of the form $$ \begin{bmatrix} 1 & c & -\frac{c^2}{2} \\ 0 & 1 & -c \\ 0 & 0 & 1 \end{bmatrix} + \begin{bmatrix} ...
- 359
2
votes
0
answers
111
views
Formal way of writing an isomorphism between unitriangular matrix groups
Let $U=U(n,R)$ the group of $n\times n$ matrices over R (a commutative unitary ring) with $1$ on the diagonal and $0$ under it. For each $0<i\leq n$ let $U_i$ be the subgroup of $U$ of all elements ...
- 634
1
vote
2
answers
43
views
$\mathbb{Z}^k/A\mathbb{Z}^k \cong\mathbb{Z}^k/A'\mathbb{Z}^k$
I am working on Ch 14 problem 4.6 from Artin's Algebra textbook.
Let $φ:\mathbb{Z}^k\to\mathbb{Z}^k$ be a homomorphism defined by $φ(x)=Ax$, where $A$ is some $k\times k$ integer matrix. Show that ...
- 1,867
0
votes
3
answers
1k
views
Prove that the map of $A \mapsto BAB^{-1}$ is an automorphism of the group of all Special Matrices $SL(\mathbb{R})$
Let $n \geqslant 1$ be an integer. Prove that for all $B \in GL_n(\mathbb{R})$, the map $A \mapsto BAB^{-1}$ is an automorphism of $SL_n(\mathbb{R})$. Where $S$ is the group of matrices with $\det = 1$...
- 255
0
votes
1
answer
150
views
$M_n(R)$ over a ring $R$
Let $R$ be a commutative ring with $1$ and $M_n(R)$ be a group of $n \times n$ matrices over $R$.
$1$. Assume that $R_1 \cong R_2$ (as rings isomorphic). What is the relation between $M_n(R_1)$ and $...
- 417
1
vote
1
answer
85
views
How to find a matrix group from an invariant form?
Sometimes I come across exam tasks that basically ask me to "Find the Matrix Group that preserves (or is isomorphic to a Group that preserves) a given function from a vector space to a field". Usually ...
- 43
1
vote
1
answer
567
views
Question on Smith normal form and isomorphism
Put
$A=\begin{pmatrix} 1 & -5 & 4\\ 1 & -2 &
13\\ -2 & 13 & 7 \end{pmatrix}.$
The smith normal form of this matrix is \begin{pmatrix} 1 & 0 & 0\\ 0 & 3 &
...
- 21
1
vote
2
answers
93
views
Show that $\varphi:\mathbb{R}→Gl_2 (\mathbb{R})$ defined by $\varphi(a)=\begin{pmatrix} 1 & a \\ 0 & 1\end{pmatrix}$ is not an isomorphism
$\varphi:\mathbb{R}→Gl_2 (\mathbb{R})$ defined by the matrix $\varphi(a)=\begin{pmatrix} 1 & a \\ 0 & 1\end{pmatrix}$
An isomorphism is a homomorphism that is also bijective. $\varphi(a)$ ...
- 87
0
votes
0
answers
73
views
Showing this a set of matrices is isomorphic to $A_4$
Let $A$ be the set of these matrices and their permutations:
$$ \begin{bmatrix}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0 \\
\end{bmatrix}, \begin{...
- 367
0
votes
1
answer
867
views
Is the group of upper triangular matrices isomorphic to $\mathbb{R}^3$?
Let
$$
G = \left\{\begin{bmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{bmatrix} : x,y,z\in\mathbb{R}\right\}
$$
I have shown that the set $G$ with the operation of matrix ...
- 459
2
votes
1
answer
458
views
Is there an isomorphism between the Kronecker Delta function and permutation matrices?
A permutation matrix is a square matrix with only a single $1$ in each row and each column, with the rest being $0$s. Here's an example:
$$K = \begin{bmatrix}
0 & 0 & 1 & 0 \\ 1 &...
- 161
0
votes
2
answers
115
views
Prove the function $f(a+bi)=\begin{bmatrix}a & b\\-b & a\end{bmatrix}$ is an isomorphism from $\mathbb{C}$ to $L$.
Let $L$ be the following subset of $M_2 (\mathbb{R})$ $$L=\left\{\begin{bmatrix}a & b\\-b & a\end{bmatrix}:a,b \in \mathbb{R}\right\}.$$
Prove the function $f(a+bi)=\begin{bmatrix}a & b\\-...
- 193
4
votes
5
answers
2k
views
Seeing $PSL_2(\mathbb{C}) \cong SO_3(\mathbb{C})$
How can I see the isomorphism between projective special linear group (order 2) and the special orthogonal group (order 3)?
I know only this setting $PSL_2(\mathbb{C}) = SL_2(\mathbb{C})/Z(SL_2(\...
- 600
2
votes
1
answer
239
views
Group isomorphism between two groups .
Let $SO_2(\mathbb{R})$ be the group of rotations of the circle under the operation of composition. And consider $\mathbb{R}$ as an additive group.
Prove that $SO_2(\mathbb{R}) \cong \begin{Bmatrix} A\...
- 2,223
1
vote
0
answers
72
views
Show $SO_2(\mathbb{R}) \cong\{A \in GL_2(\mathbb{R}) : A^tA = I, \det A = 1\}$.
Show $SO_2(\mathbb{R}) \cong\{ A \in GL_2(\mathbb{R}) : A^tA = I, \det A = 1\}$, where $SO_2(\mathbb{R})$ is the group of rotations of the circle under the operation of composition.
Attempt: Suppose ...
- 41
1
vote
0
answers
355
views
Isomorphism between matrix groups and multiplicative cyclic group of finite field
Suppose $\mathbb{F}_{p^n}$ is a finite field and $\mathbb{F}^*_{p^n}$ is the set of all non zero elements in $\mathbb{F}_{p^n}$. The set $\mathbb{F}^*_{p^n}$ is a cyclic multiplicative group. In ...
- 219
6
votes
1
answer
899
views
Order of matrix and monic irreducible polynomial over finite field
I want to verify (and prove - in case it is true) the following proposition.
Suppose $\mathbb{F}_p$ is a finite field and $m(x)$ is a monic irreducible polynomial over $\mathbb{F}_p$ with $\mathrm{...
- 219
2
votes
1
answer
587
views
$GL_k(\mathbb{Z}_p)$ is isomorphic to a subgroup of $S_{p^k}$
I would appreciate some insight. The difficult part for me is proving that the homomorphism $\Phi$ is surjective.
Consider the group action of $GL_k(\mathbb{Z}_p)$ acting on $\mathbb{(Z_p)^k}$ by ...
- 9,826
1
vote
1
answer
590
views
Are similar complex matrices again similar when each is expressed as a real matrix?
We know that, relative to this ordered basis {$(1,0),(i,0),(0,1), (0,i)$}, we can express a 2x2 complex matrix mapping $C^2 -> C^2$ as a $4x4$ real matrix (representing the same transformation of $...
- 1
0
votes
1
answer
200
views
Lie algebra homomorphisms
Let $su(N)$ denote the Lie algebra of $SU(N)$, $gl(N,R)$ that of $GL(N,R)$ and so on.
Explain why
$gl(N,R) \sim u(1) \oplus sl(N,R)$,
$ gl(N,C) \sim u(1) \oplus u(1) \oplus sl(N,C)$
$u(N) \sim u(...
- 2,870
0
votes
0
answers
89
views
For any $n>1$ , $ Aut(\mathbb Z^n) \cong GL(n , \mathbb Z)$ ? And for any $m,n >1$ , $Aut(\mathbb Z_m ^n) \cong GL(n , \mathbb Z_m)$ ?
Is it true that for any $n>1$ , $ Aut(\mathbb Z^n) \cong GL(n , \mathbb Z)$ ?
And is it true that for any $m,n >1$ , $Aut(\mathbb Z_m ^n) \cong GL(n , \mathbb Z_m)$ ?
( One problem I'm ...
user228168
3
votes
3
answers
501
views
Show that $G/H\cong\mathbb{R}^*$.
Let $G:=
\bigg\{\left( \begin{array}{ccc}
a & b \\
0 & a \\
\end{array} \right)\Bigg| a,b \in \mathbb{R},a\ne 0\bigg\}$. Let $H:=
\bigg\{\left( \begin{array}{ccc}
1 & b \\
0 & 1 \\...
- 7,617
0
votes
1
answer
60
views
Matrices within Group Theory
Recall that $GL_{2}(\mathbb{R})$ denotes the group of 2x2 invertible matrices with real entries with the product given by matrix multiplication. Let H denote the smallest subgroup of $GL_{2}(\mathbb{R}...
- 83
9
votes
2
answers
604
views
Proving that $SL_2(\mathbb{Z}_5) / \{\pm I\}\simeq A_5$
I see here that one can prove that
$$
SL_2(\mathbb{Z}_5) / \{\pm I\} \simeq A_5
$$
using the First Isomorphism Theorem.
My question is how one would do that.
I know that I need a surjective ...
- 3,389
0
votes
0
answers
83
views
Writing down elements using cycle notation.
I have $GL_2({\Bbb{Z}}_2)$ in which ${\Bbb{Z}}_2$ consists of the integers $\{0,1\}$.
We observe that $|M_2({\Bbb{Z}}_2)|=2^4$.
Now let's define Y to be the set of all non-zero elements of ${\Bbb{Z}}...
- 53