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Questions tagged [cyclic-decomposition]

Use this tag for questions about (1) in group theory, expressing a permutation in terms of its constituent cycles, (2) in commutative algebra and linear algebra, writing a finitely generated module over a principal ideal domain as the direct sum of cyclic modules and one free module, or (3) in graph theory, partitioning the vertices of a graph into subsets such that the vertices in each subset lie on a cycle.

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1answer
62 views

Let $A$ be an $n \times n$ matrix with real entries such that $A^2 + I = 0$ then $n$ is even. Not duplicated

Let $A$ be an $n \times n$ matrix with real entries such that $A^2 + I = 0$ then $n$ is even. And if $n = 2k$, then $A$ is similar over the field of real numbers to a matrix of the block form $$\...
0
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1answer
39 views

Can this diagonal system be solved by modifying it to the $Ax=B$ form?

I'd like to use the $Ax=B$ form for solving the following system. $$ \left[ \begin{matrix} t_0*d_0 & -t_1*e_0 & 0 & 0 & 0 \\ 0 & t_1*d_1 & -t_2*e_1 & 0 &...
3
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1answer
80 views

Decomposition of Cycles (Group Theory Mapping)

I have been trying to prove the following proposition below for this question. I also asked the following question here to try and get somewhere but it led to nowhere. Any help would be greatly ...
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0answers
19 views

Determining the Cyclic Decomposition of a Finitely Generated Abelian Group given a set of relations.

I am given a group $G=\text{Span}(w,x,y,z)$ with relations defined by $$\begin{bmatrix}0&0&1&3\\-2&1&1&3\\-2&4&1&3\\0&-3&1&5\end{bmatrix}\begin{bmatrix}...
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1answer
54 views

Relation between Jordan normal form and cyclic module decomposition

Let $V$ be a vector space over $\mathbb{C}$, we consider it as a $\mathbb{C}[x]$-module by choosing a linear map $\varphi:V\rightarrow V$ and for $f(x)\in\mathbb{C}[x]$ and $v\in V$, define: $$ f\cdot ...
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2answers
22 views

Need help understanding a step in a proof about modules over PIDs

This is Chapter 3, Theorem 7.3 in Algebra by W. Adkins & S. Weintraub (GTM). It's about the uniqueness of the cyclic decomposition for finitely generated modules over a PID. I highlighted the ...
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1answer
82 views

Order of a permutation in $S_n$

I'm currently in first year math and got given this question, but I have absolutely no idea how to go about it. After asking me to write the permutations given by $f$ as a product of disjoint cycles,...
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38 views

Can we characterize the set formed by cyclic vectors of a companion matrix?

Let $C$ be a matrix in companion form. Let us form a subset of general linear map $GL_n(\mathbb R)$ by \begin{align*} \mathcal E = \{G \in GL_n(\mathbb R): G = (g_1, Cg_1, \dots, Cg_1^{n-1})\}, \end{...
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1answer
35 views

Is the cyclic vector for companion matrix independent of the specific companion matrix under consideration?

Suppose $C_1, C_2 \in M_n(\mathbb R)$ are two matrices in companion form. If $v$ is a cyclic vector for $C_1$, i.e., $\{v, C_1 v, C_1^2 v, \dots, C_1^{n-1}v\}$ is a basis for $\mathbb R^n$, does this ...
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0answers
26 views

Basis of Jordan block = cyclic subspace formed by generalised eigenvectors

I think jordan matrix which are formed by jordan blocks of a particular eigenvalue , have a form of direct sum of companion matrix of a cyclic subspaces. Then basis of such cyclic subspace containg ...
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1answer
63 views

Cyclic basis for an eigenspace

Let $V=\mathbb{R}^{3}$ and $T(x,y,z)=(-3x-4y,2x+3y.-z)$. Let $\alpha=(e_{1},e_{2},e_{3})$ the usual basis for $V$. We will denote the characteristic polynomial of $T$ by $c_{T}(t)$ and the minimal ...
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0answers
55 views

How to solve $\beta^{-1}\pi^{2016}\beta = \alpha$ where $\alpha$, $\beta$ are given permutations

I am given two permutations $$\alpha = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ 5 & 6 & 7 & 8 & 3 & 11 & ...
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1answer
28 views

Relation between $G$-orbits and Cycle Decomposition of a Permutaion.

Let $X_n=\{1,2,...,n\}$, $\delta \in S_n$. Write $G=(\delta)$ and assume $G$ acts on $X_n$. What is the relation between $G$-orbits of $X_n$ and cycle decomposition of $\delta$?
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1answer
59 views

Structure Theorem for finitely generated Modules over a PID, Decomposing an Example Problem and finding Bases

I came across this Problem in Terms of my exam preparation: a.) Let N $\subset \mathbb{Z}^3$ be the submodule generated by the set {(2,4,1),(2,-1,1)}. Find a Basis {$f_1,f_2,f_3$} for $\mathbb{Z}^3$, ...
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3answers
66 views

Express a quotient of a free abelian group as a direct sum of cyclic groups

So I'm studying for my algebra quals by going over some old quals from the school. One problem asks the following. Let $G$ denote the free abelian group on $\{ x , y , z \}$ with the relations $2x + ...
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0answers
42 views

Cyclic subspaces of a linear operator with a specific minimal polynomial

Let $T:V\rightarrow V$ be a linear operator with minimal polynomial of form $m_{T} (x)= p(x)^{b}$ in which p(x) is a irreducible polynomial. Let $Z(v,T)$ denote cyclic subspace of T generated by v. ...
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0answers
89 views

How to find the cyclic vectors when finding the Rational form of a given matrix?

For a given matrix, say $A$, of dimension $n$, in $\mathbb{F}^{n\times n}$, the characteristic polynomial of $A$ is of the form $$c_A = p_1^{d_1}*...*p_k^{d_k},$$ and the minimal polynomial of the ...
2
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1answer
84 views

Calculate Smith normal form, cyclic group decomposition

Can someone please check my working for the following problem? Let $A$ be the abelian group generated by elements $x,y,z$ with relations $7x+5y+2z=0, 3x+3y=0, 13x+11y+2z=0$. Decompose $A$ as a ...
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1answer
85 views

On the intuitive idea behind decomposition into cyclic modules

Let $R$ be a ring and $M$ be a free $R$-module. If $M$ has a basis of size $n$ then we have $M \cong R^n$. I find this rather intuitive. With that as a building block, I also find rather intuitive ...
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1answer
82 views

Existence of unique maximal proper T-invariant subspace implies indecomposability

Let $V$ be a finite-dimensional vector space over a field $\mathbb{F}$ and let $T$ be a linear operator on $V$. Show that $T$ is indecomposable if and only if there is a unique maximal proper $T$-...
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0answers
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Partitioning a graph in cycles of four

I have the following question: Suppose that in a simple undirected graph with $4n$ vertices, each vertex has degree at least $2n$. Is it true that we can always partition the set of vertices ...
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0answers
56 views

How could I describe homomorphisms from $C_m \rightarrow C_n$?

E.g.: If $m = 21$ and $n = 15$. I realise I could enumerate the number of bijective mappings but I suspect I would be excluding other homomorphisms. How would I best approach this problem?
3
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1answer
73 views

How to maximize 3-cycles in this graph type inspired by block design?

I've come up with this question and I've been playing with it for a couple weeks by now without any definite breakthrough; it seems there should be a better approach than straight naive brute force, ...
5
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1answer
157 views

Alternating group $A_5$ has subgroup of order $6$ (group theory)

In my lectures, I've read that $A_5$ $($the alternating group of even length cycles in $S_5$$)$, has a subgroup of order $6$, and the example is: the group generated by $\langle (12) (34), (123)\...
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1answer
52 views

Permutation and cycles decomposition

For the permutation cycle $\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 3 & 1 & 5 & 2 \end{pmatrix}$, they decomposed it into $(3 4)(13)(45)(25)$. I don't see how ...
4
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1answer
302 views

(Average) Number of cycles of length m in permutations on N with k cycles

Suppose we have permutations on $[1,2,...,n]$ that have exactly $k$ cycles (which there are $|s(n,k)|$ of where $s(n,k)$ is the Stirling number of the first kind). What is the average number of ...
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1answer
125 views

Conjugation of permutations proof

I'm trying to prove the following statement: Let $f,g \in S_n$. Let $g = c_1c_2\dots c_k$ be the disjunct cyclus decomposition of $g$. Then the disjunct cyclus decomposition of $fgf^{-1}$ is found by ...
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2answers
416 views

How many permutations with $k$ cycles and $j$ fixed points are there?

Suppose I have the numbers $\{1,...,n\}$. I would like to know how many permutations have exactly $k$ cycles and $j$ fixed points.
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2answers
765 views

Prove or disprove: The minimum number of transpositions needed to decompose $\sigma$ is $n-S$.

Assume that $\sigma \in S_n$ and $\sigma = \alpha_1 \dots\alpha_s$ is the decomposition of $\sigma$ into disjoint cycles, such that all of the members of $\{1,2,\dots,n\}$ are appeared in the members ...
1
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1answer
55 views

If the only sub-spaces invariant under $Τ$ are $R^n$ and the zero subspace, then $U$ is diagonalizable.

Let $A$ be an $n \times n$ matrix with real entries. Let $Τ$ be the linear operator on $R^n$ which is represented by $A$ in the standard ordered basis, and let $U$ be the linear operator on $C^n$ ...
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3answers
886 views

Let $A$ be an $n \times n$ matrix with real entries such that $A^2 + I = 0$ then $n$ is even.

Let $A$ be an $n \times n$ matrix with real entries such that $A^2 + I = 0$ then $n$ is even. And if $n = 2k$, then $A$ is similar over the field of real numbers to a matrix of the block form $$\...