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} ...
1
vote
1answer
40 views

Finding invertible 3x3 matrix A such that Stab(M)A=AStab(N) for given matrices M,N

I'm reading that it is possible to find an invertible $3 \times 3$ matrix $A$ such that for the matrices $M = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 ...
3
votes
2answers
54 views

The number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$

How to count the number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$? I know that the number of invertible matrices is GL$(n,p)$. Have any ideas?
4
votes
1answer
142 views

Abelian group generators and relations

(a) Define what it means for an abelian group to be finitely generated. Explain the terms elementary divisors and rank of $G$ and describe the structure theorem for finitely generated abelian ...
1
vote
1answer
52 views

Expressing a matrix in terms of subgroup generators using Magma

With Magma, it is possible to define a subgroup $H$ of a finite matrix group $G$ in terms of generators. Given a matrix $M\in G$, Magma can also determine whether $M\in H$. Presumably, if Magma ...
0
votes
1answer
41 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 ...
0
votes
0answers
90 views

Let $ G$ be $SL_2(\mathbb{F}_5)$. Find a subgroup of $G$ isomorphic to $Q_8$, and an element of order $3$ normalizing it in $G$.

Let $G$ be $SL_2(\mathbb{F}_5)$ i.e. the special linear group of $2\times 2$ matrices $\mathbb{F}_5$. Find a subgroup of $G$ isomorphic to $Q_8$, and an element of order $3$ normalizing it in $G$. I ...
1
vote
1answer
111 views

Order of some matrices in $GL(2,p)$ is coprime with $p$

Let $M$ belongs to $GL(2,p)$ where $p$ is a prime number, and $\det M$ generate $GL(1,p)$, so I want to prove that the order of $M$ is coprime to $p$. I think if $M^{np}=I_2$ that means ...
2
votes
1answer
98 views

Prove that this group of matrices has order $p^3$

Let $G$ be a group of upper triangular matrices $\in \mathcal{M}_3 (\mathbb{Z}_p)$ with ones on the diagonal. I've already proved that this group isn't abelian, but I don't know how to show that its ...
2
votes
1answer
112 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 ...
3
votes
3answers
104 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 ...
0
votes
1answer
48 views

Non-isomorphic product of two groups

I know this is a simple question, but I'm not able to reason it out right now. Why is $\pm I_n \not\cong \pm I_n \times \pm I_n$?
4
votes
1answer
110 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 ...
3
votes
3answers
201 views

Sylow $p$-subgroups of Finite Matrix Groups

Let $G$ be a subgroup of $GL_n(\mathbb{F}_p)$, with $n\le p$, and let $P$ be a Sylow $p$-subgroup of $G$. Do all non-trivial elements of $P$ have order $p$? I believe the answer is yes, because I ...
7
votes
3answers
146 views

Problem involving permutation matrices from Michael Artin's book.

Let $p$ be the permutation $(3 4 2 1)$ of the four indices. The permutation matrix associated with it is $$ P = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 ...
7
votes
1answer
544 views

When does the adjacency or incidence matrix of a graph have consecutive ones property?

Given a graph, what are some sufficient (and necessary) conditions to tell if its adjacency matrix has the consecutive ones property? Similar question for its incidence matrix? Note that a ...
3
votes
2answers
106 views

Group does not contain any elements of order $p^2$?

I am trying to show that the group of $3 \times 3 $ upper triangular matrices over the field $ \mathbb{F}_p $ with diagonal entries 1 does not contain any elements of order $p^2$ when $ p \geq 3$. ...
6
votes
1answer
449 views

Algorithm to find conjugacy classes of subgroups/elements (in matrix groups)?

I'm looking for a simple (=doable to implement by myself) algorithm to compute the conjugacy classes of elements and subgroups of a given subgroup of $\text{P}{\Gamma}\text{L}(n,q)$. So given a group ...
1
vote
1answer
269 views

program to find matrix group given generators (Matlab)?

Given a small number of generators for a group of small (e.g. $3\times3$) matrices over a finite field $\mathbb{F}_p$ where $p$ is prime, how can we find all the elements of the group? The best I've ...
2
votes
0answers
58 views

Computing $\mathbb{C}[x,y]^G$ or $\mathbb{C}[x,y,z]^G$ where $G$ is a finite subgroup of $GL_n(\mathbb{C})$

My question is related to this link: Ring of Invariant $\mathbf{Question \;1}$. Let $$ A = \left( \begin{array}{cc} 0 & -1 \\ 1& 0 \\ \end{array} \right). $$ Then $C= \langle A\rangle$ ...
0
votes
0answers
74 views

Elementary matrix operations on group

Given a matrix whose every element is an element of group Sn (Symmetric group of n objects). I want to apply Gaussian Elimination to convert it into row echelon form. I need to find out linearly ...
5
votes
1answer
102 views

Is there a “natural” transitive action of $SL_2(\mathbb{F}_5)$ on a set with 5 elements?

I'm really looking for a "cute" way of showing that $SL_2(\mathbb{F}_5)$ is a double cover of $A_5$. The sort of action I am looking for is something like the action of $GL_2(\mathbb{F}_3)$ on ...
1
vote
2answers
195 views

Groups/Linear maps

Let $\mathbb{F}_3$ be the field with three elements. Let $n\geq 1$. How many elements do the following groups have? $\text{GL}_n(\mathbb{F}_3)$ $\text{SL}_n(\mathbb{F}_3)$ Here GL is the general ...