# Tagged Questions

64 views

### Commutator subgroup of $GL_{2}(\mathbb{Z}/p^{2}\mathbb{Z})$ is $SL_{2}(\mathbb{Z}/p^{2}\mathbb{Z})$

How would I go about showing this, where $p$ is an odd prime? The inclusion $[GL_{2}(\mathbb{Z}/p^{2}\mathbb{Z}),GL_{2}(\mathbb{Z}/p^{2}\mathbb{Z})] \subseteq SL_{2}(\mathbb{Z}/p^{2}\mathbb{Z})$ is ...
27 views

### Index of the upper triangular matrix subgroup of $SL(n,\mathbb Z)$

I want to calculate the index of the upper triangular matrix subgroup N of $SL(n,\mathbb Z)$, or how to calculate [$SL(n,\mathbb Z)$: N] is finite or infinite? Here, $n\ge 3$.
33 views

### Proving group properties of $G$, a set of $2 \times 2$ matrices with rational entries

Let $G$ be the set of all $2 \times 2$ matrices whose entries are rational numbers and whose determinant is equal to $3^n$ where $n$ is a nonnegative integer. Prove that $G$ is a group with respect ...
36 views

### What is the order of this group? [duplicate]

Let $H$ be the subgroup of the group $G$ of all $2 \times 2$ non-singular matrices whose entries are integers modulo a given prime $p$ consisting of those and only those matrices in $G$ whose ...
52 views

### Find the inverse of a matrix in $GL(2\,,\, \Bbb Z_{11})$.

What are the necessary steps and reasoning for calculating the following matrix in GL(2,$\Bbb Z_{11}$): $M = \begin{pmatrix} 2&6 \\3&5 \end{pmatrix}$. I found the answer to be ...
46 views

### How to prove that $J(M_n(R))=M_n(J(R))$?

How to prove that $J(M_n(R))=M_n(J(R))$? Here $M_n(R)$ is the ring of matrices of size $n^2$ over the ring $R$. And $J(M_n(R))$ is a two-sided ideal of the ring $M_n(R)$.
63 views

### Show that $H$ is a normal subgroup of $G$? [duplicate]

Let $\mathbb M(n;\mathbb R)$ denote the set of all real matrices (identified with $\mathbb R^{n^2}$ and endowed with its usual topology) and $GL(n;\mathbb R)$ denote the group of all invertible ...
48 views

### subgroup of $GL_{n+1}\mathbb{R}$ which is isomorphic

Describe a subgroup of $GL_{n+1}\mathbb{R}$ which is isomorphic to the group $\mathbb{R}^n$ under the operation of vector addition. I have no idea what this would look like. I would really ...
40 views

92 views

### Calculating index of a subgroup

Compute the index $[Γ( 1 ) ′ : Γ_0 ( N ) ′ ]$ where $Γ(1)' := SL(2,\mathbb{Z})$ $Γ_0(N)':= \{ \begin{pmatrix} a & b\\ c & d\\ \end{pmatrix} \in Γ(1)' : c \equiv 0 \mod{N} \}$ I'm ...
100 views

### Combinatorics inside of $GL(n,q)$

I'm studying conjugacy classes of subgroups of $GL(n,q)$ of the form $(\mathbb{Z}/p\mathbb{Z})^r$ where $q=p^d$ and $r$ is some non-negative integer. I've been able to show that for $n=p=2$ and for ...
102 views

82 views

### Finding the generators of a subgroup of $\mathrm{SL}_2(\mathbb Z)$

I am trying to solve the following problem: Let $T_{ij}(c)\in\mathrm{SL}_2(\mathbb Z)\ (i\neq j)$ be the elementary matrix which represents the elementary row operation of adding the $j$-th row ...
44 views

### Understanding a proof by R.C Lyndon and J.L Ullman.

Here in this article I have difficulties understanding the theorem on page 162. Theorem. Let $A, B$ and $C=AB$ be an elements of group $GL_2(\mathbb{Z})$, all with real fixed points. Suppose that ...
237 views

### Groups under Multiplication

Let $G=GL(2,\mathbb R)$ and $H =\left\{ \left[\begin{array}{ccc|c} a & 0 \\ 0 & b \end{array} \right]:\mbox {$a$and$b$are nonzero integers }\right \}$ under the operation matrix ...
99 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 ...
46 views

### Supremum over unitary group action

Let $A$ and $B$ are two given Hermitian operators on matrix algebra $M_n(\mathbb{C})$. $A$ is positive semi-definite with unit trace. I want to know the general method for calculating the following ...
### action of $O(n,\mathbb{R})$ on ${S}^{n-1}$
Is the action of $O(n,\mathbb{R})$ on ${S}^{n-1}$ transitive? I think this is true as orthogonal matrices are supposed to rotate and keep the length fixed, but how do I prove this? EDIT: Based on ...