# 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.

289 questions
Filter by
Sorted by
Tagged with
17 views

### Let $H$ be a subgroup of $S_n$. Show that if $H$ is a transitive subgroup of $S_n$ and $H$ is generated by some set of transpositions, then $H = S_n$ [duplicate]

So I am convinced that I am misunderstanding the question and it's probably due to the fact that I am misunderstanding the definition of $H$ being a transitive subgroup of $S_n$ Since $H$ is a ...
16 views

### Prove that a cycle of length $k\geq 2$ can be written as a product of $k-1$ transpositions.

Prove that a cycle of length $k\geq 2$ can be written as a product of $k-1$ transpositions as follows: $$(a_1 ... a_{k-1} a_{k})=(a_1 a_{k})(a_1 a_{k-1})...(a_1 a_2).$$ I found an answer here: ...
47 views

### What are the cyclic permutations on four symbols?

I am trying to solve a problem of Hamermesh's Group Theory and its Applications to Physical Problems. Problem: The cyclic permutations on four symbols form a subgroup $H$ of $S_4$. Resolve $S_4$ into ...
42 views

### Analytic estimates of limit cycle parameters

Suppose we have a two-dimensional system of differential equations, say, the well-known Van der Pol oscillator: $$\dot{x}=y, \dot{y}=\mu (1-x^2)y-x$$ Everyone knows that the study of limit cycles ...
38 views

### is there any way to find the maximal length of cycle in a permutation group?

How we can find the biggest cycle of a permutation group with given the set of generators ? is there any algorithm or theorem for that? Or how we can find the set of generators with biggest cycle as ...
15 views

### How an order of permutation can be defined?

Let $\sigma$ be the permutation: [ \begin{array}{lllllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 3 & 5 & 6 & 2 & 4 & 9 & 8 & 7 & 1 \end{...
46 views

### Cyclic order and cyclic permutation

The definitions of a cyclic permutation I know do not use the notion of a cyclic order: Definition of cyclic permutation https://en.wikipedia.org/wiki/Cyclic_permutation https://mathworld.wolfram.com/...
14 views

### How does the cycle structure of an invertible linear map over $\Bbb Z/n\Bbb Z$ compare to its affine translations?

The general question I am interested is roughly what is in the title; more concretely: Fix a positive integer $n$, and $m\in(\Bbb Z/n\Bbb Z)^\times$. For each $b\in\Bbb Z/n\Bbb Z$, define the ...
48 views

24 views

### All permutation groups acting on a finite set are closed

I'm reading through Dixon and Mortimer. Exercise 2.4.3 asks you to prove that if $|\Omega|<\infty$, then every subgroup of $\hbox{Sym}(\Omega)$ is closed. The definition of a closed permutation ...
30 views

If $g \in S_n$ and let be $1\leq k \leq n$. Prove that $g(12\cdots k)g^{-1}=(g(1)g(2)\cdots g(k))$. In $S_n$ the cycle $(12\cdots k)$ means $1\longmapsto 2$ $2\longmapsto 3$ $\vdots$ $k-1\... 0answers 13 views ### Distribution of balls into boxes. I came upon this questions on one of my tests, and wasn't able to solve it. Was wondering if anyone could point me in the right direction. Thanks in advance. Let m be the number of ways 3 identical ... 0answers 20 views ### Prove that every element of$S(\mathbb{Z})$can be written as a product of at most two elements of order 2 [duplicate] Prove that every element of$S(\mathbb{Z})$can be written as a product of at most two elements of order 2 The hint given: First show that a cyclic permutation is the product of two elements of order ... 2answers 35 views ### Simple permutation multiplication of two 4-cycles from$S_4$, specifically,$ (1324)(1423)$. For some reason I am confused on how to multiply these together with$4$elements. I can do it with$2$elements but theres a little gap in my logic and I am sure one of you can help me with a simple ... 0answers 7 views ### How does one prove that the decrement of a permutation is equivalent to the signature of the permutation The decrement$d$is defined as$n-c = d$, where$n$is the number of elements in the group and$c$is the number of independet cycles of certain permutation. If$d$is even (odd), then the claim is ... 1answer 57 views ### The Number of permutations of [n] such that$|\pi(i)-i|\leq 2$Let$a_n$be a number of permutations of the set$[n]$such that $$|\pi(i)-i|\leq 2$$. I can obtaine the GFs of these restriction permutations. It is equal $$F(z)=\frac{1-z}{1-2z-3z^3+z^5}$$ I ... 1answer 54 views ### How many ways to paint$n$windows? Given a building with$n$floors and a window on each floor. You need to paint all the windows either black, white or red. How many ways are there to paint all the windows given that there have to be ... 2answers 42 views ### Showing that$H$is a normal subroup of$A_4$Consider the subgroup$H$={$(1),(12)(34),(13)(24),(14)(23)$} How would I show that$H$is a normal subgroup of$A_4$? If I perform a few computations such as:$$(132)^{-1}(12)(34)(132)=(123)(12)(34)... 1answer 23 views ### Size of the conjugacy class of the element σ = (1, 2, 3)(4, 5) in the group Sym5 . Order of the centraliser CSym5 (σ) I am a little confused with this question. Am i right in looking for all the elements in Sym5, that are conjugate to$\sigma$, in other words$a = b\sigma b^{-1}$for$a,b \in $S5 ther will be 5 ... 1answer 31 views ### Help with permutation (13)(246)(1524) I am trying to compute a permutation:$\alpha = (13)(246)$and$\beta = (1524)$. Find$\alpha\beta$. I thought that I had a good grasp on these, but I checked it on a site, and if it is correct, I ... 2answers 445 views ### How to know if a 8 puzzle is solvable Consider the following$3\times3$sliding puzzle: This is not solvable from the following state: The explanation is that there are 11 inversions, and therefore it is unsolvable. How do they arrive ... 1answer 56 views ### Differences of Even Permutations and Latin Squares a.) Let$n$be even. Find a permutation$x_1, x_2,\dots, x_n$of the elements of$\mathbb{Z}_n$such that the differences$x_{i+1} - x_i, 1 \leq i < n$, are all different. Say if I take$n=6$, ... 2answers 36 views ### Sufficient condition that a permutation is even Let$a$be a permutation of$S_n$and suppose that$a$decomposes as a product$a = \sigma_1 \circ \dots \circ \sigma_r \circ \gamma_1 \circ \dots \circ \gamma_s$of disjoint cycles where$\sigma_i$... 1answer 91 views ### When$\langle \sigma\rangle$and$\langle\tau\rangle$intersect trivially, where both$\sigma$and$\tau$are$n$-cycles in$S_n$Let$\sigma,\tau\in S_n$be two$n$-cycles. When does$\langle\sigma\rangle\cap\langle\tau\rangle=1$? Note that$\sigma$and$\tau$are conjugate in$S_n$and WLOG we may assume$\sigma = (1,2,\dots,n)...
Given the map $\tilde{f}:[1,13]\to[1,13]$, and $\tilde{f}(x)=\begin{cases} f(x)+8;& 1\le x\le 5\\ x-8;& 9\le x \le 13\\ \end{cases}$, which looks like the figure: Prove that the map has a 10-...