# Questions tagged [permutations]

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

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### 6-letter permutations in MISSISSIPPI

How many 6-letter permutations can be formed using only the letters of the word, MISSISSIPPI? I understand the trivial case where there are no repeating letters in the word (for arranging smaller ...
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### Number of permutations of $n$ where no number $i$ is in position $i$

I am trying to figure out how many permutations exist in a set where none of the numbers equal their own position in the set; for example, $3,1,5,2,4$ is an acceptable permutation where $3,1,2,4,5$ is ...
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### Why are two permutations conjugate iff they have the same cycle structure?

I have heard that two permutations are conjugate if they have the same cyclic structure. Is there an intuitive way to understand why this is?
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### Multiplication in Permutation Groups Written in Cyclic Notation

I didn't find any good explanation how to perform multiplication on permutation group written in cyclic notation. For example, if $$a=(1\,3\,5\,2),\quad b=(2\,5\,6),\quad c=(1\,6\,3\,4),$$ then ...
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### $A_4$ has no subgroup of order $6$?

Can a kind algebraist offer an improvement to this sketch of a proof? Show that $A_4$ has no subgroup of order 6. Note, $|A_4|= 4!/2 =12$. Suppose $A_4>H, |H|=6$. Then $|A_4/H| = [A_4:H]=2$. ...
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### Find the center of the symmetry group $S_n$.

Find the center of the symmetry group $S_n$. Attempt: By definition, the center is $Z(S_n) = \{ a \in S_n : ag = ga \forall\ g \in S_n\}$. Then we know the identity $e$ is in $S_n$ since there is ...
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### Counting number of moves on a grid

Imagine a two-dimensional grid consisting of 20 points along the x-axis and 10 points along the y-axis. Suppose the origin (0,0) is in the bottom-left corner and the point (20,10) is the top-right ...
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### How to write permutations as product of disjoint cycles and transpositions

$$\sigma=\begin{pmatrix} 1 & 2 &3 & 4& 5& 6&7 &8 &9 &10 & 11 \\ 4&2&9&10&6&5&11&7&8&1&3 \end{pmatrix}$$ (1) I am ...
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### On Conjugacy Classes of Alternating Group $A_n$

In Dummit & Foote, page 131 Let $K$ be a conjugacy class and suppose that $K$ is subset of $A_n$ . Show that if $\sigma$ belongs to $S_n$ then , $\sigma$ does not commute with any odd ...
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### Number of possible permutations of n1 1's, n2 2's, n3 3's, n4 4's such that no two adjacent elements are same?

Given $n_1$ number of $1$'s, $n_2$ number of $2$'s, $n_3$ number of $3$'s, $n_4$ number of $4$'s. form a sequence using all these numbers such that two adjacent numbers should not be ...
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### Rearrangement of groups such that no two members meet again

Suppose that we are given $n$ groups of $m$ people. We want to rearrange these $nm$ people into the same format of $n$ groups of $m$ with that the catch that any two people who were originally in a ...
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### Can someone explain Cayley's Theorem step by step?

This is from Fraleigh's First Course in Abstract Algebra (page 82, Theorem 8.16) and I keep having hard time understanding its proof. I understand only until they mention the map $\lambda_x (g) = xg$. ...
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### How many ways are there for 8 men and 5 women to stand in a line so that no two women stand next to each other?

I have a homework problem in my textbook that has stumped me so far. There is a similar one to it that has not been assigned and has an answer in the back of the textbook. It reads: How many ways ...
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### How many arrangements of $\{a,2b,3c,4d, 5e\}$ have no identical consecutive letters?

How many arrangements of $\{a,2b,3c,4d, 5e\}$ have no identical consecutive letters? I find it very tough... Could anyone have some good ways?
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### “In a party people shake hands with one another”

In a party people shake hands with one another (not necessarily everyone with everyone else). (a) Show that 2 people shake hands the same no. of times. (b) Show that the number of people who shake ...
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### Does a homomorphic image of even permutations consist of even permutations?

If $f:S_n \to S_n$ is a homomorphism, prove $f(A_n) \subseteq A_n$. If every image of a transposition is even, then there is nothing to prove, but it is not sure.. How can I prove the problem?
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### number of combination in which no two red balls are adjacent.

given x spaces(you can fit 1 ball in 1 space) and unlimited number of identical red and white balls, find the total number of combinations in which no two red balls are adjacent to each other. i ...
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### Fixed points in random permutation [closed]

Suppose two random permutations of the numbers 1 to n placed side by side. a) Calculate the expectation number of fixed points for $n = 5$. b) Find the value of expectation in the amount of fixed ...
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### Existence of subgroup of order six in $A_4$

Show that the alternating group $A_4$ of all even permutations of $S_4$ does not contain a subgroup of order $6$. For me am thinking to write all elements of $A_4$ and trying to find every ...
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### Proving that any permutation in $S_n$ can be written as a product of disjoint cycles

I have attempted a proof of this, but upon looking at my notes, I feel it might be incorrect: it is noticeably simpler than the one in my notes. Proposition: any permutation in $S_n$ can be written ...
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### Arrangements of a,a,a,b,b,b,c,c,c in which no three consecutive letters are the same

Q: How many arrangements of a,a,a,b,b,b,c,c,c are there such that $\hspace{5mm}$ (i). no three consecutive letters are the same? $\hspace{5mm}$ (ii). no two consecutive letters are the same? A:(i). ...
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### The center of $A_n$ is trivial for $n \geq 4$

I need to prove that the center of $A_n$ is trivial for $n \geq 4$. $Z(A_3) = A_3$, since $A_3 = \mathbb{Z}/3\mathbb{Z}$ is commutative. One idea is two use "counting" technique. First of, all we ...