All Questions
18 questions
0
votes
1
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62
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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
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 ...
-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
$$\operatorname{Aff}(3):={\{( \begin{array}{cc}
a & b \\
0 & 1
\end{...
- 852
0
votes
2
answers
107
views
Prove $G/H \cong (\mathbb{R}-\{0\})\times(\mathbb{R}-\{0\})$
Let $G$ be the group of all real matrices of the form $\displaystyle\left( \begin{smallmatrix} a & b \\ 0 & c \end{smallmatrix} \right)$ with $ac \neq 0$ under matrix multiplication. Let $H$ ...
- 555
1
vote
0
answers
32
views
Is $H/L$ isomorphic to $K^*\times K^*$ where $K^*=(K_{\ne0},\cdot)$ with $K$ field and $H, L$ certain subgroups of $GL(2,K)$?
Let $K$ be a field, $H=\left\{\begin{bmatrix}a&b\\0&d\end{bmatrix}:a,b,d\in K, ad\ne0\right\}$, $L=\left\{\begin{bmatrix}1&b\\0&1\end{bmatrix}:b\in K\right\}.$
I'm asked to prove that ...
- 472
5
votes
2
answers
892
views
If S is a normal subgroup, identify the quotient group G/S. What are the $\varphi(G)$'s?
The following is an exercise from Artin's Algebra: (Kiefer Sutherland's voice)
Let G be the group of upper triangular real matrices $\begin{bmatrix}
a & b\\
0 & d
\end{bmatrix}$ with a and ...
user198044
1
vote
2
answers
43
views
$\mathbb{Z}^k/A\mathbb{Z}^k \cong\mathbb{Z}^k/A'\mathbb{Z}^k$
I am working on Ch 14 problem 4.6 from Artin's Algebra textbook.
Let $φ:\mathbb{Z}^k\to\mathbb{Z}^k$ be a homomorphism defined by $φ(x)=Ax$, where $A$ is some $k\times k$ integer matrix. Show that ...
- 1,867
0
votes
3
answers
1k
views
Prove that the map of $A \mapsto BAB^{-1}$ is an automorphism of the group of all Special Matrices $SL(\mathbb{R})$
Let $n \geqslant 1$ be an integer. Prove that for all $B \in GL_n(\mathbb{R})$, the map $A \mapsto BAB^{-1}$ is an automorphism of $SL_n(\mathbb{R})$. Where $S$ is the group of matrices with $\det = 1$...
- 255
0
votes
1
answer
867
views
Is the group of upper triangular matrices isomorphic to $\mathbb{R}^3$?
Let
$$
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
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
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
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
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}}...
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