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

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How to calculate no. of binary strings containing substring “11011”?

I need to calculate no of possible substrings containing "11011" as a substring. I know the length of the binary string. Eg: for a string of length 6, possible substrings are: 110110 110111 111011 ...
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1answer
18 views

# of DISTINCT sequences of numbers $a_1, …, a_n$ (sometimes $a_i=a_{i+1}$)

As from the title, given a sequence of number $a_1, ..., a_n$ I would like to know how many number of different sequences exist taking into account that same numbers might exist. Examples: f(0003) ...
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27 views

Affine geometry and its basis

I am trying to solve an exercise from the book "Permutation Groups" by J. Dixon and B. Mortimer, but, this is not a homework. The affine geometry $AG_d(F)$ consists of points and affine subspaces ...
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37 views

If $G$ acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $N \unlhd G$. Then every element outside of $N$ fixes at most $p$ $N$-orbits.

Let $G$ be a transitive permutation group acting on $\Omega$ such that each nontrivial element either fixes no point or exactly $p$ points for some prime $p$. Also assume that for $g \notin ...
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1answer
27 views

The kernel of an action on the orbits of normal subgroup if group acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$

Let $G$ be a permutation group acting transitively on $\Omega$ and suppose $N \unlhd G$ is a normal subgroup of $G$. Assume that for $g \in N_g(G_{\alpha})$ we have $$ G_{\alpha} \cap G_{\alpha}^g = ...
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3answers
50 views

Comprehension: Consider the $8$ digit number $N=22234000$

Comprehension: Consider the $8$ digit number $N=22234000$. $(1)$ How many possible $8$ digit numbers can be formed using all $8$ digits of $N$? $(a)$ ${8\choose 3}.\frac{5!}{2!}$ $(b)$ ...
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1answer
30 views

In how many ways can three men and five women (all distinguishable) line up for a group if the people at each end must be opposite sex?

I think it must look like this W _ _ _ _ _ _ M and M _ _ _ _ _ _ W with the middle not mattering. Would it be 5 ways for the first spot * 3 ways for the last spot * 6! and then the opposite for the ...
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1answer
22 views

Does isomorphism transfer the transitative property between permutation groups?

If $G$ is isomorphic to $G'$ and $G$ is transitive to $S_n$ then does it not immediately follow that $G'$ is also transitive to $S_n$? Do I need to state some results or theorems to prove this or is ...
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36 views

Permutation on a strange string

There is a strange string of 10 characters ether '0' or '1'. I have n filter strings each having 10 characters ether '0' or '1'. A '1' at the i-th position in a filter string means that if I applies ...
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1answer
33 views

The kernel of an action on blocks, specifically the action on the orbits of normal subgroup

Let $G$ be a permutation group acting transitively on some set $\Omega$ and suppose we have a normal subgroup $N \unlhd G$. Then the orbits of $N$ form a system of blocks, and if $\Delta$ is such an ...
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1answer
23 views

Prove the group transitivity of alternating group $A_n \quad n>2$?

Does it not suffice to point out that $$(i, k)(i, j)\in A_n$$ The element at location $i$ is mapped to the element at location $j$ and and the element at location $j$ is mapped to some third ...
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1answer
47 views

How many shapes are possible from gluing together the faces of n cubes?

Say I have n cubes. I am allowed to glue the faces of these cubes together, but the faces must line up perfectly. How many unique shapes could I make? All orientations of one shape are considered to ...
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0answers
26 views

If $G$ acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $N \unlhd G$. Then $G_{\alpha}N$ is normal if we have $p$ orbits of $N$.

Let $G$ be a transitive permutation group acting on $\Omega$ such that each nontrivial element either has no fixed point or exactly $p$ fixed points. Suppose that for $g \notin N_G(G_{\alpha})$ we ...
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1answer
23 views

Need help answer checking combinations and permutations problem

I was doing some problems for my quiz earlier today (which is now concluded) and went through some combination problems I'm unsure I answered correctly. If I'm wrong, can someone please explain why to ...
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0answers
35 views

If $G$ is solvable and acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $M$ is maximal normal. Then $|G/M| = p$.

Let $G$ be a transitive permutation group on $\Omega$ which fulfills the following property (P) (P) each nontrivial element fixes no point or exactly $p$ points. for some odd prime $p$. Further ...
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1answer
25 views

Is $Y(y)=xy$ a permutation?

My book says that a permutation is a bijective mapping from a group to itself. So, let our group be $S$ and let our function be $Y(y)=xy$ where $x,y\in S$. Now we know that since $S$ is a group, $xy ...
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29 views

What does it mean that the Frobenius representation of a group is unique, and what are its consequences

For a Frobenius group its kernel is a characteristic and nilpotent group, the last property restricts the possibilities how a given group could be represented as a Frobenius group. A statement of this ...
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1answer
27 views

If $G$ is solvable and acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$, then maximal normal subgroups are Frobenius groups

Let $p$ be an odd prime and let $G$ be a solvable, transitive permutation group such that each nontrivial element fixes no points or exactly $p$ points on a set $\Omega$. Further suppose that for $g ...
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1answer
39 views

Extended Josephus permutations generated by keyword

The (well known) generalized Josephus algorithm consists in starting from the ordered set $Z_n=\{1,2,...,n\}$, and choosing and removing cyclically from left to right each m-th element until the set ...
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0answers
30 views

If $G$ acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $N \unlhd G$. Then $G$ acts on the $N$-orbits in the same way.

Let $G$ be a finite transitive permutation group on $\Omega$ such that every nontrivial element either fixes no point of $\Omega$ or fixes exactly $p$ points of $\Omega$. Assume that for $g \notin ...
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14 views

Ball Stacking Permutation Debate

We have a child's toy (FP Stack and Roll Ball Set). There are 10 half dome cups that can be stacked in a variety of ways. It was hotly debated among our group how many ways to stack them are ...
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1answer
37 views

How does a permutation $P$ affect the singular value $\sigma_{\text{max}}(Q^\top P^\top Q)$ for orthogonal $Q$?

Let $q_i$ for $i=1,\ldots,m$ be the columns of the matrix $Q\in\mathbb{R}^{n\times m}$, $n\geq 2m$, which are pairwise orthonormal ( i.e. $q_i^\top q_j = \begin{cases} 1 & \text{if}\quad i=j \\ 0 ...
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42 views

Is there a name for this generalization of the determinant?

In the context of averaging over network paths, I arrived at a certain generalization of the determinant for an $n\times n$ square matrix $A$, that is $$D_k(A) := \sum_{(j_1,j_2,...,j_n):\,\, ...
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2answers
225 views

Find all three digit 'special' numbers which equal the average of all permutations of their digits [closed]

The number $518$ has a special feature. Lets make the six permutations of this number and add them together ($158+185+518+581+815+851=3108$). This number's average is $\dfrac{3108}{6}=518$. How can ...
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2answers
31 views

What does it mean to say a single element acts semi-regularly

Let $G$ be a group acting on some set $\Omega$. Just a minor point, but saying that some nontrivial element $g \in G$ acts semi-regularly, does this mean that $g$ itself has no fixed point, or that ...
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2answers
20 views

What is $D_{10}/\left \langle \sigma \right \rangle$ where $D_{10}$ is the Dihedral Group or a regular Pentagon? [duplicate]

$D_{10} = ({1, \sigma, \sigma^2, \sigma^3, \sigma^4, \tau, \sigma\tau, \sigma^2\tau, \sigma^3\tau, \sigma^4\tau}) $ where $\sigma = (12345)$ and $\tau = (13)(45)$. I'm stuck on trying to calculate ...
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1answer
26 views

Distinct birthdays problem. Verification of solution.

Question Consider $n$ people who are attending a party. We assume that every person has an equal probability of being born on every day of the year, independent of everyone else. Assuming that nobody ...
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2answers
351 views

In how many ways can we arrange 4 letters of the word “ENGINE”?

I need to know a combinatoric solution to this problem, with Generating functions in book, gives us 102. It might be a very simple problem, but Im very confused with this. Would be very nice of you ...
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2answers
118 views

Combinatorics Choosing Objects Under Condition

If 28 objects are arranged in a circle at equal distance from each other, in how many ways can 3 objects be chosen such that no two are adjacent or diametrically opposite.
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45 views

set of permutations of set $\lbrace 1,2,..,n \rbrace$ [duplicate]

Prove the following lemma for set of permutations of $\lbrace 1,2,\dots,n \rbrace$ $(n\geq2)$ and a fixed number $k\neq1$: $\textbf{lemma: }$The number of permutations where 1 and $k$ are in the same ...
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1answer
12 views

Enumeration problem where we have two consecutive events happening

Two experiments are to be performed. The first can result in any one of $m$ possible outcomes. If the first experiment results in outcome $i$, then the second experiment can result in any of $n_i$ ...
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31 views

If $P\unlhd G$ is semiregular and $G$ such that $|\mbox{fix}(g)| \le 2$ for $g \ne 1$. Each $g \ne 1$ has at most one fixed point in each $P$-orbit

Let $G$ be a finite group acting nonregular and transitive on $\Omega$ such that each nontrivial element has at most two fixed points and $|\Omega| \ge 4$. Suppose that $N$ is a proper normal ...
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1answer
31 views

If $G$ acts nonregular, transitive and $|\mbox{fix}(g)| \le 2$, $|G_{\alpha}|$ is odd and $|\Omega|$ is even, then $|G|$ has twice odd order

Let $G$ act nonregular and transitive on $\Omega$ such that each nontrivial element has at most two fixed points. Let $\alpha, \beta\in \Omega$ be distinct and such that $U := G_{\alpha}\cap ...
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1answer
52 views

Is this proof of $x^0=1$ correct?

We know that, $$^n P_r = \dfrac{n!}{(n-r)!}$$. We also know that $^n P_r$ is the number of ways to arrange $n$ objects in $r$ places. Now if repetition is allowed then the number of ways to arrange ...
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1answer
76 views

Question about symmetric group action

I am trying to analyse the following. Assume $S_{3}$ acts on a non empty set $T$, and that is has $3$ orbits. What can we say about the possible cardinalities for the set T? My thoughts: If ...
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1answer
29 views

Arrange distinct groups of balls

We have 10 green balls and 7 white balls. What should be estimated, is the number of the ways they can be arranged if those balls are distinct with no consecutive white balls. Is my approach, that ...
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2answers
72 views

Group Theory: How to find all possible images of $ f $?

Let $H$ be a group and suppose that $ f: D_{10} \rightarrow H $ is a homomorphism. How do I describe and justify all the possible images of $f$. $D_{10} = ({1, \sigma, \sigma^2, \sigma^3, \sigma^4, ...
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49 views

If $G$ acts such that $|\mbox{fix}(g)|\le 2$ for $g \ne 1$ and $O_p(G) \ne 1$ and $|G_{\alpha}|$ odd. Assertions about Frobenius groups

Let $G$ be a finite group acting nonregularly and transitive on $\Omega$ such that each nontrivial element has at most two fixed points and $|\Omega| \ge 4$. I know three facts: i) If $1 \ne X \le ...
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0answers
21 views

For a specific subgroup $N$ of index $2$ why $( S \setminus Q ) \cap N = \emptyset$ if $S \in \mbox{Syl}_2(G)$ and $|S : Q| = 2$

Let $G$ be a finite group acting on some set $\Omega$ with $|\Omega|$ even. Let $S \in \mbox{Syl}_2(G)$. Further let $Q \le S$ such that $|S : Q| = 2$ and suppose we have some $x \in S \setminus Q$. ...
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1answer
15 views

Is it possible to find all subgroup of $Alt_4$ from all subgroup of the permutation group $S_4$?

Is it possible to find all subgroup of $Alt_4$ from all subgroup of the permutation group $S_4$? I think the answer is yes from Lagrange's theorem and Sylow's theorem. Is anyone is able to give ...
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1answer
54 views

Spivak's curious thoughts about the action of permutations.

Here is an excerpt of Spivak's Differential Geometry. What I do not understand is why he believes $\sigma \cdot (\rho \cdot v) = (\rho\sigma) \cdot v$. Since $\sigma$ and $\rho$ are elements of ...
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3answers
87 views

Random permutation two sets

We have got two sets $A = \{1, \ldots, k \}$ and $B = \{k, \ldots, 2 k - 1\}$. Let $\sigma$ be random permutation of $\{1, \ldots, n\}$, $n \ge 2k - 1$. Let $A_{max} := max\ \sigma(A)$, $B_{min} := ...
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1answer
22 views

What segment in 8-bit LED Displays used for Traffic Light timers can be removed causing minimal impact in the readability of the countdown numbers?

I passed by an intersection with traffic lights and noticed that 1 segment of the 8-bit display counter is dimmed (it's not working). When the lowermost segment is dimmed for example, number 4 can ...
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1answer
19 views

Number of ways to select a president and a secretary with restrictions

Solve using addition principle: A committee composed of Jesse, Bianca, Ray, and Lily is to select a president and a secretary. How many selections are there in which Jesse is president or not an ...
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0answers
32 views

For a group acting such that each nontrivial element has at most two fixed point, size of orbit of single $2$-power order element

Let $G$ be a nonregular, transitive permutation group on $\Omega$ such that each nontrivial element has at most two fixed points. Suppose $S \in \mbox{Syl}_2(G)$ and that we have $\alpha, \beta \in ...
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2answers
24 views

Can someone explain what this statement means (Groups and permutations) please?

I'm currently reading notes on a lecture I missed due to not feeling well. These are notes on "Symmetric Groups and Modular Arithmetic Groups". A sentence in the notes says: "For a set $S$, a ...
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1answer
24 views

Can I place the rubiks3cube pieces in the distorted position I intend to get?

Right now I am trying to get the distorted position like this: in each face only one diagonal is solved and no similar colour is on a face other than the diagonal pieces mentioned previously. For ...
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1answer
47 views

Prove, by logical reasoning, rather than by formula, the following permutation identities

The formula would have been useful but I am not really good at logical reasoning especially in permutations so I need help from you guys to identify errors in my answer (as well as give hints for part ...
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0answers
28 views

Describe the centralizer

Exercise from Artin's Algebra. Describe the centralizer $Z (\sigma) $ of the permutation $\sigma = (153)(246)$ in the symmetric group $S_{7} $, and compute the orders of $Z (\sigma) $ and of $C ...
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35 views

How to generate the next permutation?

I feel confuse with the next permutation. We have $4$ step for find the next permutation. step $1$: find from right to left and check $a[i] < a[i+1]$. step $2$: find the pivot with condition is ...