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Questions tagged [permutation-cycles]

For elementary questions concerning permutation cycles and permutation groups. This includes all representations of permutations (two-line arrays, cycles, bipartite graphs); transpositions and the sign/parity of a permutation; the Symmetric group and the Alternating group. To be used with the (permutations) tag to make the distinction between abstract algebra permutation questions and combinatoric permutation questions.

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permutations cycle

I am doing abstract algebra problems, but unfortunately, the book we are using for the class is quite poor and leaves out lots of definitions and explanations, so I am not even sure if I completely ...
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Number of times we have to compose a permutation in order to have exactly k fixed points

Let $f = (1 4 6)(2 7 5 8 10)(3 9)$ in $S_{10}$. Find an integer $n$ such that $f^n$ has exactly $7$ fixed points. I provided the exact numbers, but would welcome a more general solution.
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the order of a product two elements in symmetric group.

Let $G$ be a group, let $a,b\in G$ with $ab=ba$, and let $|a|=m,~|b|=n$. Then, $|ab|\mid\textrm{lcm}(m,n)=mn/d$, where $d=\gcd(m,n)$. I have confusing in the following situation: Let $\sigma,\...
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Confusing the order of two product of permutations [duplicate]

Let $\sigma$ and $\tau$ be distinct(means relative prime) permutations of order $m$ and $n$, respectively(need not be $\gcd(m,n)=1$) in certain symmetric group. Now, suppose that $\sigma\circ\tau=\...
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2answers
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Count the number of permutations of certain cycles type

Suppose we have $n$ elements, assume there is a permutations over $k$ elements among the $n$ elements so $n-k$ are fixed. Let that the permutation over the k elements is represented by permutation ...
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Permutations in $S_9$ with length $4,3,2$.

How many permutations in $S_9$ have one cycle of length $4$, one of length $3$, and one of length $2$? My attempt so far is: ${9 \choose 4}*{5 \choose 3}*{2 \choose 2}$, which is the way to choose ...
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Confused about composition of two permutation cycles

I'm slightly confused about the product of the following permutation cycles. I am given that $s_1 = (1\ 2)$ and $s_2 = (2\ 3)$ where both are generators for the symmetric group $S_3$. My textbook ...
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The permutation is given. For how many functions $f:\Bbb{N_{10}} \rightarrow \Bbb{N_{10}}$ are $f(\pi(i))=\pi(f(i))$?

The permutation $\pi\in S_{10}$ is given by the table: \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline i & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \pi &...
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Question regarding Symmetric group and induction [duplicate]

I have no clue what to do here ! Let $n>1$ be a natural number. the m-cycle $\sigma=(a_1,a_2,a_3,\dots,a_m)$ denotes a permutation in $S_n$ where $\sigma(a_1)=a_2,\sigma(a_2)=a_3,\dots, \...
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Proving $\phi$ is a homomorphism and that the group G is simple.

Let $G,H$ be groups where $H$ is a subgroup of $G$ and $[G:H] = 3$, let $g_{1},g_{2},g_{3}$ be the elements of the group $G$. Let $c_{1} = g_{1}H,c_{2} = g_{2}H,c_{3} = g_{3}H$ be three distinct left ...
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Compute the product of cycles that are permutations of $S_8$ [duplicate]

Here is the product of cycles. My textbook goes right to left multiplication. $$(1,4,5)(7,8)(2,5,7)$$ An observation here is that I don't think these cycles are disjoint and therefore they are not ...
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number of permutation of S4 as product of two disjoint cycles each of length 2

There was a problem of finding out the number of permutations of order 2 in S4. There are two cases. case-1 permutation of single cycle of length 2. case-2 permutations of two disjoint cycles ...
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1answer
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Composition of permutations in cycle notation

For $p=(1\ 4\ 3\ 2)$, find $p^2$. The textbook states that the solution is $p^2=(1\ 3)(2\ 4)$. Now I understand that $1 \mapsto 4 \mapsto 3$, and $2 \mapsto 1 \mapsto 4$, however, why must the result ...
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Possibilities for a certain subgroup of $S_n$.

A two-part problem: $\bullet$ Let $H \leq S_n$ be the subgroup that fixes 1. Show that $H$ is isomorphic to $S_{n-1}$. $\bullet$ Show that there are no proper subgroups of $S_n$ that properly ...
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Presentations of subgroups of S6 as permutations

Computer science professor, self-taught abstract algebraist. Beginner with GAP and SAGE. Can someone show me the quickest way to, when given the description of a subgroup of S6, obtain its ...
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Finding the automorphism group of a graph

Let me denote the graph in the picture by $\Gamma$ and I will refer to the points as numbers $1-9$. I need to find Aut($\Gamma$). Looking at this graph, it seems that there can be a permutation on ...
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Permutation of 6 girls and 15 boys in a circle following a specific rule

How many ways are there to arrange 6 girls and 15 boys in a circle such that there are at least 2 boys between any 2 adjacent girls? Please help me how to proceed with this. My approach: Arrange ...
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Prove that a permutation is a $p$-cycle

I have difficulties proving the following theorem: Let $σ ∈ S_n$ satisfy $σ \neq (1)$, and $σ^p = (1)$, where p is a prime number such that $\frac{n}{2} < p ≤ n$. Prove that σ is a p-cycle. ...
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1answer
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Prove or disprove: $S_{10} = \langle (1,3),(1,2, … ,10) \rangle$

Prove or disprove: $S_{10} = \langle (1,3),(1,2, ... ,10) \rangle$ I know that $S_{10}=\langle (1,2) , (1,2,...,10) \rangle$. I tried to use this fact to prove the above but failed. It made me think ...
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How to get an expression for this isomorphism

Define $\hat{\mathbb{R}} = \mathbb{R}\cup\{{\infty}\}$ and $1/0=\infty,1/\infty=0, \infty/\infty=1,1-\infty=\infty-1=\infty$. Then the six functions $\hat{\mathbb{R}}\to\hat{\mathbb{R}}$ given by $x$, ...
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2answers
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Describing the normalizer of a cycle in $S_{10}$

Let $\sigma = (1,2,3,...,9) \in S_{10}\;$ a $10$-cycle in the permutation group $S_{10}$ a. What is the size of $N_{S_{10}}(\langle \sigma \rangle)$, The normalizer of the group generated by $\sigma$ ...
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Counting inversions of random elements in coxeter groups

I am trying to find a general interperetation to the following facts (pls be patient to read it). Let's look at the property of Kendall-Mann numbers $M(n)$ which are row maxima of Triangle of ...
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Product of cycles in symmetric group S7

I have this question and although I don't want the answer directly, how do I go about finding other ways of expressing the product of two cycles in S7? Thanks
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Round-robin tournament scheduling where no team plays twice in a row, for n teams games

Inspired by this question here:, I would conjecture that so long as there are 2n+1 teams involved in a round-robin tournament where each games consists of n-way teams, then a schedule is possible ...
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1answer
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Chain Of Transpositions Raised To A Power And Its Orbits

Suppose we have a set of $n$ elements which we would like to permute. Let $\begin{pmatrix} i & j\end{pmatrix}$ denote the transposition (the swapping) of elements $i$ and $j$. I would like to ...
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Find $f^{−100}g^{146}f^{301}$ (permutations to high powers)

Find $f^{-100}g^{146}f^{301}$ where $$f = \begin{pmatrix} 1 & 2 & 3& 4 & 5 & 6 & 7 \\ 3 & 1 & 5 & 7 & 2 & 6 & 4\end{pmatrix}, \\ g = \begin{pmatrix} 1 &...
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Let $\sigma \in S_n$ be a cycle. Prove that $\sigma$ can be written as the product of at most $n-1$ transpositions.

I know that $$(a_1,a_2,...,a_n)=(a_1a_n)(a_1a_{n-1})...(a_1a_3)(a_1a_2)$$ But how to prove this decomposition has the maximum number of non-repeating transpositions? One start point may be that this ...
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Determine the number of cycles of the form $(\underline{6})(\underline{2})(\underline{2})$ in $S_9$

Consider a product of cycles of the form $(\underline{6})(\underline{2})(\underline{2})$ in $S_9$ where $(\underline{i})$ is a cycle with length $i$. I want to determine the number of such cycles. ...
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1answer
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Restoring permutation from differences of adjacent elements

Suppose a permutation $\pi \in S_n$ is encoded by a list of integers $P=(p_1, p_2, ... p_{n-1})$, where $p_i = \pi(i+1) - \pi(i)$, i.e. $P$ is the list of differences of adjacent elements. Now, given $...
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How to calculate the Galois group of $x^5+15x+12$?

How to calculate the Galois group of $x^5+15x+12$ over the field $\Bbb Q$? Using the Tchebotarov Density theorem which states that "the density of primes $p$ for which $f(x)$ splits into type $T$ ...
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1answer
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Writing $2$-cycles as a product of adjacent 2-cycles.

Problem: Consider the arbitrary 2-cycle $(a\ b)$ from $S_n$. Find a way to write this permutation as a product of adjacent 2-cycles. What I do know: A transposition is a single cycle of length 2. An ...
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Number of different cyclic permutations in a symmetric group

So a permutation cycle is written as $(a,b,c,d)$ where $a\to b\to c\to d\to a$, I would like to know the number of permutation cycles I can express $(a,b,c,d)$ such that the permutation cycles are ...
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Generating random permutation with N cycle

Suppose we have integer list from 1 to N. I am interested in generating random permutations such that its cycle notation has only one N cycle. What algorithms are known to generate random N-cycle ...
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Find the number of $6 \times 7$ matrices with entries $[0,1]$ such that their row and column sums are odd. [closed]

My attempt is quite handwaivy. But I think this has something to do with permutation matrices. I am absolutely new to this topic. can anyone throw any light on this solution? I know there are $2^{n\...
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1answer
60 views

Find permutations whose third power is known

I have to find permutations $a$ such that $a^3=(1 \ 2)(3 \ 4)(5 \ 6)(7 \ 8 \ 9 \ 10)$ and I have to find at least 3 solutions. So first I must find disjoint cycles. Those are: ...
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Determine if exists a subgroup of order $3$ of $H=\langle\sigma^{8440}\rangle$

Consider the following permutation of $S_{13}$. $\sigma=(1\;3\;13\;5\;11\;8)(2\;10\;4\;6\;12\;7\;9)$ Determine if exists a subgroup of order 3 of $H=\langle\sigma^{8440}\rangle$. If yes, exhibit it ...
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Multiplying Permutation Groups in Cycle Notation

I am given that f = (1 2 3 4 5 6) and g = (1 2). I am told to compute $f^2gf^4$. I know that $f^2 = (1 3 5)(2 4 6)$ and $f^4 = (1 5 3)(2 6 4)$. But I am having trouble computing the product of all ...
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Permutation group equation: $x^{20} = \sigma$

I need some help solving the next permutations group equation: $$ x^{20} = \sigma $$ where $$ \sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & ...
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How many solutions does $x$ have in $x^3 = \sigma^6$ in $S_{7}$

Let $\sigma=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 2 & 3 & 5 & 7 & 1 & 4 & 6\end{pmatrix} \in S_{7}.$ How many solutions does $x^3 = \sigma^6$...
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1answer
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A question about cardinality of differences of subsets of $\{1,\cdots, n\}$

Let $n\in\mathbb{N}$ and for every $1\leq i\leq n$, let $A_i=\{1,\cdots, i\}$. For every sequence of distinct numbers $i_1,\cdots,i_k\in\mathbb{N}$, let $$M(i_1,\cdots,i_k)=(A_{i_1}\setminus A_{i_2})...
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1answer
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Sylow 2 subgroups of S4

I am trying to find all the Sylow 2 subgroups of S4 using Sylow’s theorems. Now, I know that a Sylow 2 subgroup of S4 has size 8, and that there are either 1 or 3 of them (as the number of of Sylow 2-...
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1answer
35 views

Permutation and permutation matrices

When we want to define transformations using permutations, what are the subtle differencies betwen the use of permutation matrices, and the use of permutations? Say I want to define a way to shuffle ...
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0answers
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Proof of Determinant property via definition in linear algebra

Let A=$(a_{ij})_n$ be a square matrix & B=$(b_{ij})_n$ be another square matrix obtained after interchanging two rows of A, say $p^{th}$ and $q^{th}$. Then |B|=-|A|. Proof Given:- We have, $b_{...
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2answers
38 views

Compositions & Transpositions of permutations

Consider the set of all permutations $S_n$. Fix an element $\tau\in S_n$. Then the sets $\{\sigma\circ\tau\mid \sigma\in S_n\}= \{\tau \circ\sigma\mid \sigma\in S_n\}$ have exactly $n!$ elements. ...
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1answer
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Given a cycle $c \in S_n $ with $ ord(c) = s $ and $ s = kt $, prove that $c^k$ is a product of $k$ cycles of length $t$. [closed]

I came across this question in a recent exam. Given that $ ord(c) = s $, we assume that $c^s = c^{kt} = (id) \implies (c^{k})^t = (id)$. That means that $c^k$ is a cycle of order $t$. Can you ...
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1answer
34 views

On terminology; what is meaning of the “decrement” of a permuation? (or what is the alternative word or phrase used for this definition)

I am currently studying some lecture notes in Russian and I frequently look up the theorems and topics in English for better understanding (I'm not a native Russian so I merely just use the Russian ...
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2answers
74 views

In how many ways can 3 employees visit 40 locations

Three employees need to visit 40 different cities under the following conditions: each location should be visited by exactly one employee, and no location should be visited multiple times. The travel ...
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1answer
29 views

Permutation as string position recording in Wilson's FSG book

I'm reading Wilson's The Finite Simple Group book, mainly the Alternating group chapter, which explains the splitting criterion quite in detail. While heading there, I came across a possible ...
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1answer
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On the product of two cycles (and its conjugates)

So this is from Charles C. Pinter's "A Book of Abstract Algebra"- specifically, it's from the second chapter on permutations. The question is: Let $\alpha_1$ and $\alpha_2$ be cycles of the same ...
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1answer
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Geometrical meaning of A4 conjugacy classes of elements of order 3

I know that elements of order 3 in A4 group split into two separate conjugacy classes, due to lack of odd elements to form a unique class like happens in the full S4. Since A4 is the rotation ...