# 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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### Proving $\frac{a}{b^3}+\frac{b}{c^3}+\frac{c}{a^3}\geqslant \frac{a+b}{b^3+c^3}+\frac{b+c}{c^3+a^3}+\frac{c+a}{a^3+b^3}$

For $a,b,c>0.$ Prove$:$ $$\dfrac{a}{b^3}+\dfrac{b}{c^3}+\dfrac{c}{a^3}\geqslant \dfrac{a+b}{b^3+c^3}+\dfrac{b+c}{c^3+a^3}+\dfrac{c+a}{a^3+b^3}\quad (\text{Tran Quoc Thinh})$$ It's easy with ...
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### Proving $\sum_{\text{cyc}}\, (23a-5b-c)(a-b)^2(a+b-3c)^2 \geqslant 0$

Problem (KaiRain's problem). For $a,b,c\geqslant 0.$ Prove $$\displaystyle \sum_{\text{cyc}}\, (23a-5b-c)(a-b)^2(a+b-3c)^2 \geqslant 0$$ I only found a proof by $pqr.$ (Note that from pqr's proof we ...
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### How many $g \in S_6$ are there such that $gxg^{-1} =x?$

If $x = (123)(456)$ and $y = (234)(561)$, count the number of $g \in S_6$ such that $1. gxg^{-1}=x$; and $2. gxg^{-1}=y$ For 1, I got that $g = e_{S_6}$ (the identity in $S_6$) is one of them, ...
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### Question about a line of reasoning with cyclic subspaces

The following reasoning was used in a proof of the cyclic decomposition theorem by Hoffman and Kunze (for reference, part 3 of the proof, page 235). Suppose $T$ is a linear transformation, and I have ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### (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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### 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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### 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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### 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 ...
### 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 $C^n$ ...
### 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 \...