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

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5
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2answers
125 views

Rubik's Revenge Cube in GAP

I'm trying to create the Rubik's Revenge (4x4x4 cube) group in GAP . Take the following net of the 4x4x4 cube with each sticker labelled with a number. The front, left, upper, right, down, and back ...
2
votes
2answers
41 views

Isomorphic subalgebras of $K^n$ are just “permutations” of each other?

Let $K$ be a field$^{[1]}$ and $n$ be a positive integer. The $K$-vector space $K^n$ endowed with pointwise multiplication is an unital commutative associative $K$-algebra. Suppose $A$ and $B$ are ...
0
votes
1answer
23 views

Confused about the number of permutations of the Enigma Machine

I recently learned about the Enigma Machine in my cryptography class, but I am a bit confused as to the number of permutations of the wheel settings. According to every article I've read on the ...
1
vote
1answer
50 views

Find the left cosets of subroups of $S_3$

So I am struggling to understand the definition of a coset. If I have the following symmetric group $S_3=\{1, \sigma, \sigma\tau, \sigma\tau^2, \tau, \tau^2\}$, where ...
1
vote
0answers
30 views

Colored beads on a loop

Suppose we have $p$ beads of $n$ different colors on a loop. $p$ is a prime number and we consider the loop to be the same if one is a rotation of the other. Then how many distinct beads are there? By ...
1
vote
1answer
36 views

General formula to compute the exponent of the symmetric group $S_n$

Someone has already asked whether an exponent less than $n!$ is possible for a symmetric group $S_n$. It has been answered that it is for $n \ge 4$. I would like to know if there is a general ...
0
votes
2answers
28 views

8-puzzle which has the numbers in order but has gap in between

If the numbers of the 8-puzzle are all in order, but the blank tile is somewhere in between, is this puzzle solvable? (for example, $\begin{bmatrix} 1 & 0 & 2 \\ 3 & 4 & 5 \\ 6 & 7 ...
1
vote
0answers
45 views

Number of permutations with two elements in one cycle

Show, that for a set of permutations of a set $\{1,\dots,n\}$ $(n>0)$ the following statement is true. statement: The number of permutations where $1$ is in the same cycle with $k$, and the number ...
0
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1answer
46 views

Expression of the number of distinct necklaces with $m$ beads of $n$ different colors.(rotations don't count)

Suppose I want to form an $m$-bead necklace such that each bead can have $n$ different colors. Necklaces which differ by a rotation are considered the same and beads of the same color are ...
0
votes
1answer
28 views

Number of cycles and sign

For an even $n \in \mathbb{N}$, the sign of $\tau \in S_n$ is 1 if and only if the number of disjoint cycles in $\tau$ is even? And it is 1 if and only if the number of disjoint cycles in $\tau$ is ...
2
votes
1answer
44 views

Sign of permutations and cycle lengths

For any $\tau \in S_n$, the sign of $\tau$ is 1 if $\sum_{c\in\tau}((\text{cycle length of } c) -1)$ is even; the sign of $\tau$ is $-1$ if $\sum_{c\in\tau}((\text{cycle length of } c) -1)$ is odd? ...
0
votes
0answers
32 views

If $G$ acts so that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$. Conditions such $S \in \mbox{Syl}_p(G)$ has maximal class.

Let $G$ be a nonregular, transitive permutation group acting on $\Omega$ such that each nontrivial element either fixes no point or exactly $p$ points for some prime $p$. Further suppose that for $g ...
12
votes
4answers
219 views

Prove that $(mn)!$ is divisible by $(n!)\cdot(m!)^n$

Prove that $$(n!)\cdot(m!)^n|(mn)!$$ I can prove it using Legendre's Formula, but I have to use the lemma that $$ ...
2
votes
1answer
35 views

How many solutions does the equation $\sum_{i=1}^{k}{x_i}=c$ have, given that the $x_i\in\mathbb{Z}$ and $0\leq x_i\leq d$?

We are given initially some $k,c,d\in\mathbb{N}$. How many solutions $(x_1, x_2, ..., x_k)$ does the equation $\sum_{i=1}^{k}{x_i}=c$ have, where $x_i\in\mathbb{Z}$ and $0\leq x_i\leq d$?
0
votes
2answers
60 views

Basic Combinatorics: How many sequences have at least 3 red balls?

Pick a sequence of 10 balls from a sack containing red, blue, green, yellow, white and black balls. Each time a ball is picked, it is replaced in the sack before the next ball is picked. a) How many ...
0
votes
1answer
35 views

Counting problem involving permutations - verification please?

I have tried solving this, but I'm unsure if I'm right. Any suggestion would be appreciated. Question: 15 kids arrive at camp and are assigned a place to sleep. There are 3 different cabins each of ...
0
votes
0answers
17 views

If $G$ is solvable and acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $M$ maximal, why is $N_M(M_{\alpha}) \in \mbox{Syl}_p(M)$?

Let $p$ be an odd prime. Suppose $G$ is solvable and acts as a nonregular and transitive permutation group on $\Omega$ such that each nontrivial element either fixes no point or exactly $p$ points. ...
1
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0answers
36 views

If $G$ acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$, $M$ maximal with $|G : M| = p$, then $|M / L| = p$ for semiregular $L \unlhd G$.

Let $G$ be a solvable, nonregular and transitive permutation group acting on $\Omega$ such that each nontrivial element either fixes no point or exactly $p$ points for some prime $p$. And suppose that ...
2
votes
1answer
30 views

Why is this formula for counting property of bit strings true?

Consider the set $S$ (whose elements are indexed by $i\in\{1,...,\binom{l}{m}\}$) of bit strings of length $l$ that contain exactly $m$ $1$s and $l-m$ $0$s. For each string $s_i\in S$, define ...
0
votes
1answer
31 views

Let $g \in N_G(H)$ an element of order $5$. Compute the order of $H \langle g \rangle$.

Let $G = A_5$ and $H=\langle(1,2,3,4,5)\rangle$. Let $g \in N_G(H)$ an element of order $5$. Compute the order of $H \langle g \rangle$. I think I can use the second isomorphism theorem to solve ...
4
votes
4answers
97 views

Details about Caley's Group Theorem

The Caley-group-theorem states that every group is isomorphic to a subgroup of a permutation group. I am especially interested in the case that the group is finite. My question : If G is a group ...
0
votes
4answers
36 views

How to multiply in Sym(X) [duplicate]

Could someone show me how to multiply in say $S_4$. I know how to multiply say $(4321)(2341)$ but when it comes to ones that do not contain $4$ terms, like $(34)(231)$, I have no idea how to handle ...
-1
votes
2answers
64 views

How many ways we can arrange 4 letters from PROFESSOR? [duplicate]

How many ways we can arrange 4 letters from PROFESSOR? The way I tried to do is by grouping the repeated words like (OO), (RR), (SS) and that can be done in three ways which seems to me a bit ...
0
votes
2answers
19 views

Minimizing over permutations

Fix $a=(a_1,...,a_n)^T,b=(b_1,...b_n)^T \in \mathbb{R}^n$. Assume WLOG $a_1\geq...\geq a_n$, $b_1 \geq ... \geq b_n$. Let $s$ be a permutation of the indices $\{ 1,...,n \}$. Intuitively, the way ...
0
votes
1answer
19 views

Sum of all numbers formed by the given digits

The formula for sum of all numbers formed with all the given digits is: (Sum of digits) (n-1)!(1111....ntimes) n stands for number of digits. For ex: Sum of all ...
1
vote
1answer
67 views

Which algorithm GAP uses to check equality of two groups?

How can i check which Algorithm is used by GAP for its working. Like ...
0
votes
4answers
75 views

Rearrange letters in “ENGINE” so no letter appears next to itself

How many ways are there to arrange the letters in "engine" so that no letter appears next to itself? Initially, I know that there are 6! possible arrangements of the letters. But we have to divide ...
3
votes
2answers
76 views

Checking whether a list of Permutations form a Group

I am new to Group Theory and GAP as well. I am given a list of Permutations say gap> a:=[(1,2), ()] How can i check whether these permutations form a Group themselves. Apart from the obvious method ...
0
votes
2answers
26 views

Father Buys Nine Different Toys

Came across another textbook question I'm struggling with... A father buys nine different toys for his four children. In how many ways can he give one child three toys and the remaining three ...
2
votes
1answer
20 views

Different Face Dice - Permutations and Combinations

If four fair dice are tossed, what is the probability that they will show four different faces? Ok so I get that the answer is (1/6)^4 x 6P2, but am wondering why it is a permutation and not a ...
2
votes
3answers
49 views

What type of problem is this? Combinatorics?

Given 10 cups and 8 non-distinct balls, how many ways can we distribute the balls among the cups such that no cup has more than 2 balls in it? Cups are allowed to be empty, as required by the problem ...
1
vote
1answer
12 views

Cycle type of elements in left regular presentation

I am interested in the proof of the following statement: Let $G$ be a finite group, and let $\pi : G \to S_{\vert G \vert}$ be the left regular presentation. If $x \in G$ has order $n$, and $\vert G ...
1
vote
2answers
39 views

What is formula for obtaining this?

I'm new to this part of SE. I do not even know what I should put as a title (hopeful someone can help me edit). Here is my problem I have a bus traveling from A to D (trip). On the way there is ...
0
votes
1answer
20 views

Permutation/Combination problem - choosing 2 objects from different categories

Context: there are three candies in a basket (5 red, 4 blue, 5 green). How many ways can you choose 2 candies with different color? Should I use permutation or combination w/ this? I tried using: ...
0
votes
2answers
18 views

Permutations and Combinations - Disc101

In how many ways can 32 people walk through 7 doors? My attempt :- C(32,7) But on the test I got this answer incorrect so can anybody help me figure out the actual answer?
1
vote
2answers
54 views

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 ...
0
votes
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) ...
1
vote
0answers
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 ...
0
votes
2answers
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 ...
2
votes
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 = ...
1
vote
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 $N$? $(a)$ ${8\choose 3}.\frac{5!}{2!}$ $(b)$ ...
1
vote
1answer
29 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 ...
0
votes
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 ...
0
votes
0answers
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 ...
1
vote
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 ...
2
votes
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 ...
3
votes
1answer
46 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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
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 ...