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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.

Cyclic decomposition is used in three areas of mathematics:

In group theory, cyclic decomposition is a useful convention for expressing a permutation in terms of its constituent cycles. One can use cyclic notation to express a permutation as a product of cycles corresponding to the orbits of the permutation; that distinct orbits are disjoint leads to the terminology decomposition into disjoint cycles.

In commutative algebra and linear algebra, cyclic decomposition refers to writing a finitely generated module over a principal ideal domain as the direct sum of cyclic modules and one free module.

In graph theory, a cycle decomposition is a decomposition (a partitioning of a graph's edges) into cycles. Every vertex in a graph that has a cycle decomposition must have even degree.

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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 of $\alpha_1, \dots, \alpha_s$. Is this…
Paolo Molina
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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 $$\begin{bmatrix} 0 & -I \\ I & 0…
User8976
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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 in $n$ parts of size $4$ such that the vertices of every…
digital-Ink
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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 $\langle (12) (34), (123)\rangle$. I don't even understand what group we are…
Nina
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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 Buffalo Way and computer so I will not post it. (Please…
NKellira
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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 can get SOS but very ugly and my SOS solution is…
NKellira
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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{align*} where the first column $g_1$ of $G$ is a…
user1101010
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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 cycles of length $m$ for one of these permutations?…
videlity
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$\text{rank }\begin{bmatrix} A - \lambda I & b \end{bmatrix} = n$ for every eigenvalue $\lambda$ implies $b$ is cyclic.

The Problem. If $A$ is a complex $n \times n$ matrix and $b$ is a complex column vector and if $\text{rank }\begin{bmatrix} A - \lambda I & b \end{bmatrix} = n$ for every eigenvalue $\lambda$ of $A$, then $b$ is cyclic. My Attempt. I know that…
1Teaches2Learn
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When is an integer power of a $12$-cycle a $12$-cycle?

I'm given a $12$ cycle $\sigma = (1\dots 12)$, and want to find for which $i$ is $\sigma^i$ also a $12$-cycle. We know that if $G$ is a group and $g \in G$, with $|g| = n$, then $|g^a| = \frac{n}{(n,a)}$, so in this case I want to say that $i$ must…
Irving Rabin
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How does $(abc) = (ac)(ab)$?

I know that permutations in the symmetric group, permutations are the finite products of transpositions. This is given: $$ (abc) = (ac)(ab) \\ (abcd) = (ad)(ac)(ab) \\ \vdots \\ (a_1a_2 \cdots a_k) = (a_1a_k)(a_1a_{k-1})\cdots(a_1a_2) $$ However,…
daikons
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Writing permutations as products of adjacent transposition.

I want to write $(1,2,4,3)$ as a product of adjacent transpositions, i.e., transpositions of the form $(k \;\; k +1)$. Well, I manage to change this cycle to $(1,3)(1,4)(1,2)$, and $(1,3)=(1,2)(2,3)(1,2)$, $(1,4)=(1,2)(2,3)(3,4)(2,3)(1,2)$. Hence,…
Chang Henry
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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 appreciated! Proposition: Let $\sigma\in S_{n+k}$.…
W. G.
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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 that given some relations that the generators $x_1,…
RGS
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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, but I haven't come up with anything conclusive…
Wildcard
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