# 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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### Some characterizations about T-indecomposable vector spaces

Recently I have found this statement and I would like to prove as exercise. Let $V$ be a vector space with $dim_{\mathbb{K}}(V)=n \in \mathbb{N}$ and $T\in End(V)$. The following facts are equivalent:...
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### An exercise about cyclic subspace

I'am learning linear algebra and have encountered this problem Let $V$ be a finite - dimensional $\mathbb{C}-space$, $A$ be a linear operator on it,$W\subseteq V$ be a cyclic subspace of $V$. Prove ...
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### Can every finitely generated module be written as a sum of simple modules?

Let $R$ be a ring with unity. Take for granted that an $R$-module $M$ is simple if and only if it is cyclic, and that if $m \in M$ then $Rm$ is a submodule of $M$. Let $M$ be an $R$-module and ...
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### Decompose ${\mathbb{Z}_n}^{*} / ({\mathbb{Z}_n}^{*})^2$ into a direct sum of cyclic groups. [duplicate]

Decompose ${\mathbb{Z}_n}^{*} / ({\mathbb{Z}_n}^{*})^2$ into a direct sum of cyclic groups, Where $({\mathbb{Z}_n}^{*})^2 = \left\{a^2 \ : \ a \in {\mathbb{Z}_n}^{*} \right\}$, A) when $n$ is an odd ...
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### Why is a hamming (7,4) coding scheme symmetric?

I'm trying to understand the explanation on pages 2/5 of the file and 18 in the book http://www.inference.org.uk/mackay/itprnn/ps/17.21.pdf on why the (7,4) Hamming code is symmetric. The explanation ...
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### 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 ...
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### 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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### 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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### 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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### 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 $n$ ...
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### 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?