For questions related to permutations, which can be viewed as re-ordering a collection of objects.

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How many cycles $A$ and $B$ can form this cycle

How many cycles $A$ and $B$ can form this cycle: $AB=(axyguimjrcwk)(bvqphsleofzt)(d)(n)$ I can see that $A$ and $B$ must share the cycle $(dn)$, and I believe due to ordering, both $A$ and $B$ must ...
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1answer
21 views

$H$-orbits in X have not the same cardinality if $H$ is not normal in $G$

Let $G$ be a transitive subgroup of the symmetric group $S_n$ on $n$ letters, and let $H$ be a normal subgroup of $G$. I know that the action of $G$ on the set $X =\{ 1,..., n \}$ induces a natural ...
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2answers
10 views

consecutive combination of n things taken k

Let's start my question with a simple example. Suppose I have $4$ apples that are numbered 1 to 4 and I want to to choose $2$ of them. The first and easy answer is $4C2$ but I want them to be ...
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1answer
27 views

Is this permutation: s^(-1) ◦ t^(-1) ◦ s ◦ t even or odd? [on hold]

Is this permutation: $s^{-1} \circ t^{-1} \circ s \circ t$ even or odd?
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3answers
40 views

Prove that $Z(S_n) = \{(1)\}$ for every $n \geq 3$. Induction

I wonder if this questions can be done by induction. $S_3 = \{(1),(12),(13),(23),(123),(132)\}$ $Z(S_3)$ contains all the elements in $S_3$ that commutes with all the element in $S_3$ We can easily ...
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2answers
23 views

Writing permutations as products of disjoint cycles

How can I write these permutations as products of disjoint cycles? i.$\;\;(1234)(513)$ ii.$\;\;(13526)(53)(46215)$ iii.$\;(13)(12)(32)(143)$
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Naively showing that $A_n$ mod a nontrivial normal subgroup is abelian.

Suppose $H \lhd A_n$ is a nontrivial normal subgroup of the alternating group on $n$ letters. Without using the fact that $A_n$ is simple, prove that $A_n/H$ is abelian. Can this be done? I will ...
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1answer
17 views

Problem in permutation groups involving conjugates

I have to find a permutation $a$ satisfying $ a xa^{-1}=y$ where $ x=(12) (34)$ and $y=(56) (13)$ My attempt in solving the problem was- $$ a(12)(34)a^{-1}= a(12)(a^{-1}a)(34)a^{-1}= ...
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1answer
26 views

a problem involving permutation groups

I am given two cycles $(123)$ and $(456)$ and have to find a permutation $ \sigma$ (if it exists) such that $ \sigma(123) \sigma^{-1} = (456)$. This is what I tried: let $ \sigma(123) \sigma^{-1} ...
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2answers
75 views

Permutations and Combinations of infinite cases [on hold]

A deck of $52$ cards labeled from $1$ to $52$ is shuffled randomly. You do the following 2 steps: reveal the first card on the top of the deck Slot it into a first book labelled "$1$" Then you ...
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23 views

Clarification on two combination problems [on hold]

So I have two questions that I was hoping someone could explain to me what they are asking in layman's terms, and maybe some advice on where to start with them. Thanks for any help! 1)Let S and T be ...
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1answer
17 views

Probability of an event if items are drawn simultaneously or not

A box contains 10 balls, with each ball labeled from 1 to 5 (there are two balls labeled with a 1, two labeled with a 2, and so on). 3 are drawn without replacement. What is the probability of drawing ...
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2answers
57 views

Counting all possibilities that contain a substring

How many strings are there of seven lowercase letters that have the substring tr in them? So I am having a little problem with this question, I know that the total number of combinations is $26^6$ ...
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2answers
24 views

Take seven courses out of 20 with requirement

To fulfill the requirements for a certain degree, a student can choose to take any 7 out of a list of 20 courses, with the constraint that at least 1 of 7 courses must be a statistics course. ...
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0answers
46 views

Am I over counting?

Two chess players, A and B are going to play 7 games. Each game has three possible outcomes: a win for A (which is a loss for B), a draw (tie), and a loss for A (which is a win for B). A win is ...
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2answers
44 views

What is the answer to this P&C problem.

What is the answer to the below mentioned P&C problem: BurgerTown offers many options for customizing a burger. There are 3 types of meats and 7 condiments: lettuce, tomatoes, pickles, ...
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0answers
28 views

All non-isomorphic transitive actions of the Dihedral group

Consider the Dihedral group $D_n$ of order $2n$ as a permutation group. That is $$D_n = \langle (1,2,\ldots, n), (1)(2, n-1)(3,n-2) \cdots \rangle.$$ I would like to determine all faithful transitive ...
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1answer
31 views

Permutations: Discrete Math

How many permutations are there of the set $(a,b,c,d,e,f,g)$ My Answer: Since there are 7 elements in the set, $7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$ Am I right?
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1answer
26 views

Permutation Game problem

could you help me to understand this problem? This is the problem statement Alice and Bob are playing a game called "The Permutation Game". The game is parameterized with the int N. At the start of ...
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1answer
30 views

Determining the size of an automorphism group for a given design

I'm trying to wrap my head around the idea of automorphisms, and I'm having a lot of issues. One of the questions I've been given as an exercise is thus; Let $\mathbb{V} = \{1, 2, 3, 4, 5, 6\}$ ...
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3answers
40 views

$S_4 \ne \langle (1,2,3,4), \, (1,3)\rangle$

So I'm trying to prove $S_4≠⟨(1,2,3,4),(1,3)⟩$, and I get the basic idea that $(1,2)$ swaps two things next to each other, which neither of the other operations do, and necessarily neither do their ...
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1answer
46 views

Finite groups acting on strings.

Let $s = abcdandsoon.. \ \in \Sigma^*$. Let $|s| = n$ be the length of $s$. Consider all permutations of the positioned symbols that make up $s$, such that $s$ is fixed under the permutation. So if ...
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1answer
26 views

Question related to permutations and combinations [on hold]

How many numbers greater than $1000$, but not greater than $4000$ can be formed with the digits $0,1,2,3,4$, repetition of digits being allowed ? Please explain every step . The answer in my book is ...
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1answer
68 views

How does the Enigma machine ensure that no letter is substituted for itself?

In Alan Turing: The Enigma Andrew Hodges describes how the letter encodings performed by a German Enigma machine "would always be swappings" (original emphasis). And goes on to say that There was ...
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2answers
35 views

What are the transitive groups of degree $4$?

How can I find all of the transitive groups of degree $4$ (i.e. the subgroups $H$ of $S_4$, such that for every $1 \leq i, j \leq 4$ there is $\sigma \in H$, such that $\sigma(i) = j$)? I know that ...
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0answers
14 views

can someone explain what is a permutation pattern class?

I understand that involvement is having a subset of a set \alpha order-isomorphic to set \beta. A pattern class is a set of permutations closed under taking subpermutations. what does that ...
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2answers
37 views

Ways to arrange 4 different colour balls with no two of the same colour next to each other

I have n green balls, n blue balls, m red balls, m yellow balls. How many ways are there to arrange this such that we don't have an sequence with 2 of the same colour next to each other? I don't ...
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42 views

Solution to $x(128)x=(12365)(479)$ in $A_9$, the alternating group

This isn't homework. I'm wondering if anyone knows techniques besides trial and error to find $x$ or show there is no solution, to problems like this, or worse, say $xpx^2rx^{-4}=s$, where $p$, $q$, ...
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2answers
39 views

Number of ways to re-arrange INTERNATIONAL with L to the right of E.

How many ways can you re-arrange I N T E R N A T I O N A L such that L is always to the right of E, it does not need to be a specific number of places, so both E L I N T R N A T I O N A and I N ...
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37 views

Why is the permutation $(a,c,d,e)(a,b)=(a,d)(b,c,e)$

I'm working through a proof in my notes. We already know that the transposition $(a,b)\in G$ and $(a,b,c,d,e)\in G$, where $G$ is a group of permutations of the elements $a,b,c,d,e$, so it's a ...
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Family of four and eight people ordered around a table

So the question is how many ways can a family of four which includes the mother, father and two children, be ordered around a table with eight other people if the mother must sit beside the father and ...
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1answer
37 views

Automorphisms of B_n

Consider the Coxeter group of type $B_n$. This group, of order $2^n n!$, can be identified with the group of odd permutations of the set $\{\pm 1,\dots,\pm n\}$ and is thus isomorphic to the ...
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Permutations with repeating digits

My question is this : how many distinct two digit numbers can be produced from numbers $4, 3, 3, 1$? When applying the formula $$\frac{4!}{(4-2)!2!}$$ you come up with $6$, yet when doing the problem ...
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2answers
57 views

Outer automorphism of $S_6$ and conjugate stabilizers

Let $f:S_6 \mapsto S_6$ be an outer automorphism of $S_6$ and consider the subgroups $$G = \{\pi \in S_6 \mid \pi(1) = 1\}$$ and $$H = \{\pi \in S_6 \mid f(\pi)(1) = 1\}.$$ I would like to show that ...
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110 views

Elements of order 10 in S6

I conjecture that there are no elements of order 10 in $S_{6}$. My reasoning is that in order for there to exist an element of order 10, then it must be a cycle of order 10 or it's a composition of ...
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44 views

Number of order permutations

Can someone explain how a set {1,2,3,.....,n} has n!/6 many permutations where 1 is to the left of 2 and 3, and 2 is to the left of 3. This works for all the n I tested but I can't make sense of why. ...
3
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2answers
119 views

permutations confusion!

Hello this is my first post , I am reading a book called (probability for dummies) the answer in the book for the question below has confused me ... Suppose you have four friends named Jim , Arun , ...
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3answers
68 views

permutation & combinations

How many odd three digit numbers are there when tens digit is greater than units digit and hundreds digit is greater than tens digit? $225$ $ 45$ $ 50$ $230$ My attempt: The units digit can be ...
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1answer
51 views

Applications of the following theorem in the real world

We know that every permutation can be expressed as a product of transpositions ( cycles with length 2). As a class project I'm looking for the applications of this fact in the real world; especially ...
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3answers
44 views

Stochastic variable exercise: People between me and my friend.

This is the exercise: $n$ people are arraged randomly in a line (not a circle), among which are yourself and a friend. Call $Y$ the number of people that are between you and your friend. Show: $E[Y] ...
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1answer
166 views

How many combinations can 18 teams be put into a ladder from 1st to last [closed]

I'd like to know individual combinations 18 teams will be made placing the teams first to last. Also following from that, if I said three teams would not be put in the bottom three, and another ...
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1answer
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Abstract Algebra Symmetric Groups

$$ \begin{align} \beta &= \begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 1 & 3 & 8 & 7 & 6 & 5 & 2 & 4 \end{bmatrix} \\ &= ...
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Permutations with fixed points

I am trying to write a java program that counts permutations of a string, I would like to check my results by hand, but I can remember (or find) the formula to count the number of permutations, and ...
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List all possible subgroups of $A_4.$ Determine which subgroups of $A_4$ are normal.

I have a question which is List all possible subgroups of $A_4$. Determine which subgroups of $A_4$ are normal. Since $|A_4| = 12,$ the order of any proper subgroup of $A_4$ must be an element of ...
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1answer
30 views

Determine the number of ways for n couples to stand in a line so that no one stands beside his or partner (explanation for the answer)

I'm not quite sure if I'm understanding solution to following problem: "Determine the number of ways for n couples to stand in a line so that no one stands beside his or her partner." The general ...
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Find eight elements in S6 that commute with (12)(34)(56). Do they form a subgroup of S6?

I know the question has been asked and answered many times, but I am trying to shore up my understanding of this concept. Given the questions here and here, does this mean that I could rearrange the ...
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2answers
29 views

Interpretation of a Problem involving permutations

[USAMO 1999 submission, Titu Andreescu] Let $n$ be an odd integer greater than $1$. Find the number of permutations $p$ of the set $\{ 1, 2, …, n\}$ for which $$\def\x#1{\lvert p(#1)-#1\rvert} ...
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473 views

How many teams can be made from 11 people?

The question asks this: Five places exist on a team. $11$ total people. $6$ come from district A, $4$ from district B and $1$ from district C. How many different groups of five are there? How many ...
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Group theoretic construction for permutation algorithm

Consider a permutation $\sigma = [s_1, \ldots, s_n]$. The `contracting endpoints' construction for the subsequence $[s_i,\ldots, s_k]$ is given by iteratively taking the product of cycles given by the ...
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Equivalence relation among matrices

Consider the set of $p\times q$ matrices with entries from the set $S=\{1,\dots,s\}$. Say that two such matrices are equivalent if one can be transformed into the other by a series of operations of ...