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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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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
29 views

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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2answers
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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
45 views

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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2answers
36 views

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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1answer
20 views

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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39 views

Find the number of $6 \times 7$ matrices with entries $[0,1]$ such that their row and column sums are odd.

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
57 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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1answer
36 views

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
51 views

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
117 views

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
30 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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46 views

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
37 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
27 views

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
30 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
72 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
26 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
43 views

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
34 views

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 ...
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1answer
31 views

Proving the sign map is a homomorphism

Definition: A transposition is a 2-cycle permutation. Definition: A permutation $\sigma$ is called even if its decomposition into transpositions has even number of transpositions; analogously for odd ...
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1answer
56 views

The centralizer of a cycle is the group generated by the elements of $S_n$ that are disjoint from the cycle and the group generated by the cycle.

I am having a hard time trying to prove that the centralizer of a cycle is the group generated by the elements of $S_n$ that are disjoint from the cycle and the group generated by the cycle. It is ...
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1answer
314 views

Number of homomorphisms between two arbitrary groups

How many homomorphisms are there from A5 to S4 ? This is how I tried to solve it. If there is a homomorphism from A5 to S4 , then order of element of S4 should divide the order of its preimage. Now ...
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1answer
18 views

commutativity of cycle permutations [closed]

How is the following true, (given two cycle permutations a and b) $(a)^{-1} (b)^{-1} = ((b)(a))^{-1}$ where b and a contain one of the same elements. isn't it only disjoint cycles that are commutative?...
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Explanation of an unexpected observation in modular arithmetic

Let the multiplication graph $n:m$ be the graph with $m$ points equally distributed on a circle and a line between points $a$ and $b$ when $n\cdot a \equiv b\operatorname{mod} m$. Looking at the ...
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1answer
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Alternate definition of symmetric group action [duplicate]

Let $S_3$ act on $\Re^3$ by the permutation matrices P(σ), where P(σ) permutes the basis vectors: $$P(σ)(e_i) = e_{σ(i)}$$ Show that, with this formula, $$\sigma(x_1,x_2,x_3) = P(\sigma)(x_1e_1 + ...
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3answers
54 views

Let $n$ be an odd positive integer and $a\in S_n$ be an $n$-cycle. Show that the order of $C(a)$ must be odd.

I am working on the following problem from group theory: If $n$ is odd and $a\in S_n$ is an $n$-cycle, $a=(a_1,a_2,......,a_n)$, show that no element of the centralizer $C(a)=\{g\in S_n \mid ga=ag\}...
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1answer
37 views

Find the conjugacy class $O_{(234)}$ if $G=A_{4}$ is acting on itself by conjugation.

Find the conjugacy class $O_{(234)}$ if $G=A_{4}$ is acting on itself by conjugation. I calculated $(x)(234)$ for all $x\in A_{4}$ and got the set $O_{(234)}=\left \{(234), (143), (142), (123), (132)\...
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1answer
20 views

Compute all $X_{g}$ and all $G_{x}$ for $X = \left \{1, 2, 3\right \}$, $G = S_{3} = \left \{(1), (12), (13), (23), (123), (132)\right \}$.

Compute all $X_{g}$ and all $G_{x}$ for $X = \left \{1, 2, 3\right \}$, $G = S_{3} = \left \{(1), (12), (13), (23), (123), (132)\right \}$. Can someone give me a head start to this problem? $X_{g}=\...
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1answer
23 views

Order and sign of a cycle

I would usually show workings or attempts of a method when posting a question here, but for this I am completely lost. I don't believe it to be a very hard question but it is as follows: Determine ...
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2answers
72 views

Cycle structure of the permutation $x \mapsto p·x\operatorname{mod}q$ for coprime $p,q$

Let $[q] = \{0,\dots,q-1\}$, $p < q$. Consider the function $\mathbf{p}: [q] \rightarrow [q]$ which sends $x \mapsto p·x\operatorname{mod}q$, i.e. the multiplication by $p$ modulo $q$ on $[q]$. ...
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1answer
86 views

Permutations and terminology

Say I have the following permutation $$\sigma ={\begin{pmatrix}1&2&3&4&5&6&7&8\\1&2&3&8&4&5&6&7\end{pmatrix}}$$ which consists to let ...
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1answer
20 views

find all $x \in S_5$ such that $x^3 = (12)$

So I have to find all $x \in S_5$ such that $x^3 = (12)$. For example, one solution would be $(12)$ itself, because its order is $2$. How can I find all of the solutions though? Is it just trial and ...
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1answer
53 views

Express (1,…,n) as a product of 2-cycles

In Alan F. Beardon's "Algebra and geometry" he asks in an exercise to express $(1\ \ldots\ n)$ as a product of two cycles: Show that $(1\ 2\ 3\ 4)=(1\ 4)(1\ 3)(1\ 2)$. Express $(1\ 2\ 3\ 4\ 5)$ as ...
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1answer
31 views

Definition of a permutation as product of cycles

I am having trouble with the following passage of "Algebra and Geometry" by Alan F. Beardon in the highlighted part. I've read it several times but I do not understand what he is saying. Can you give ...
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185 views

Passing a binary message to a friend where one of the components is always “turned on”

Suppose I want to communicate an integer to a friend between 1 and 1000. In order to pass this message I use a $k$-vector whose entries can be set to $0$ or $1$. So for example if $k=3$, a natural way ...
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(Permutations) For $n ≥ 2$,there are at least n subgroups of Sn of order $(n − 1)!$

For $n ≥ 2$, show that there are at least n subgroups of Sn of order $(n − 1)!$
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Number of derangements using PIE

The statement of Principle of Inclusion and Exclusion that I have been given is $card(X$ \ $(\bigcup\limits_{i=1}^{\infty} A_{i}))=\sum\limits_{I\subseteq[n]}(-1)^{card(I)}card(A_I)$ $A_I=\...
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How many cuboids can be created using 125 unit cubes

A $5\times 5\times 5$ cube is built using unit cubes.How many different cuboids(that differ in at least one unit cube) can be formed using the same number of unit cubes. My Attempt: Let $x_{i}$ units,...
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1answer
23 views

Show that $\tau (18632)(47) \tau^{-1} = (12345)(67)$.

Example 4.7. of Aluffi's Algebra says In $S_8$, $(18632)(47)$ and $(12345)(67)$ must be conjugate since they have the same type. So there there exist $\tau$ such that $\tau (18632)(47) \tau^{-1} = (...
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1answer
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Combinatorics: Number of permutations from $1$ to $60$, with cyclic permutation of length $l > 30$

In a game, $60$ kids have the chance to win presents. The organiser gives every kid a number between $1$ and $60$, so that all kids have different numbers. In a room, there is a cupboard with $60$ ...
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How to prove that an even permutation of $A_n$ is a square of another permutation from $S_n$?

I am trying to go through a proof which contains a statement that an even permutation from $A_n$ is a square of another permutation from $S_n$. My basic ideas are like this: Suppose an even ...
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1answer
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Prove: two elements in the group of permutations satisfy A^2 = B^2 then A = B.

How does one prove when two cycles in the group of permutations satisfy A^2 = B^2 that A = B? I think the fact the inverses A_*A = e and A_ = A^(l-1) and A = A^(l+1) is useful. l is the length of ...
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How is the permutation relationship interpreted

I was given the example as an illustration of structure of permutations in my lecture notes on algebra as shown below: $\bigl(\begin{smallmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 &...
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All permutations carrying the set {1,2} into the set {1,2}

Book question: Describe the following subgroups of $S_4$ and determine the number of elements in each subgroup: a) All permutations carrying the set {1,2} into the set {1,2} My question: ...
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Help understanding the permutation group action .

I'm trying to understand actions in regards to group theory . specifically in my notes I found the following example : Say G=$A_4$, for $x \in \{1,2,3,4\}$, and $\tau \in A_4$ We let $x^{\tau}$ be ...
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1answer
44 views

Existence of a permutation consisting of exactly two cycles for odd prime

Is it true that if p is an odd prime number, then there exists some integer a such that the permutation of Ф(p) induced by multiplication by a mod p consists of exactly 2 cycles?