Tagged Questions

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

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How many 7-note musical scales are possible within the 12-note system?

This combinatorial question has a musical motivation, which I provide below using as little musical jargon as I can. But first, I'll present a purely mathematical formulation for those not interested ...
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Why is the determinant defined in terms of permutations?

Where does the definition of the determinant come from, and is the definition in terms of permutations the first and basic one? What is the deep reason for giving such a definition in terms of ...
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What is the shortest string that contains all permutations of an alphabet?

What is the shortest string $S$ over an alphabet of size $n$, such that every permutation of the alphabet is a substring of $S$? I thought of this problem while reading a open problem on shortest ...
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Is War necessarily finite?

War is an cardgame played by children and drunk college students which involves no strategic choices on either side. The outcome is determined by the dealing of the cards. These are the rules. A ...
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How many lists of 100 numbers (1 to 10 only) add to 700?

Each number is from one to ten inclusive only. There are $100$ numbers in the ordered list. The total must be $700$. How many such lists? Note: if, as it happens, this is one of those math problems ...
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Six Frogs - Puzzle (order reversal using special transpositions)

I had come across a puzzle: The six educated frogs in the illustration are trained to reverse their order, so that their numbers shall read 6, 5, 4, 3, 2, 1, with the blank square in its ...
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Can $G≅H$ and $G≇H$ in two different views?

Can $G≅H$ and $G≇H$ in two different views? We have two isomorphic groups $G$ and $H$, so $G≅H$ as groups and suppose that they act on a same finite set, say $\Omega$. Can we see $G≇H$ as permutation ...
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“Efficient version” of Cayley's Theorem in Group Theory

I'm considering finite groups only. Cayley's theorem says the a group $G$ is isomorphic to a subgroup of $S_{|G|}$. I think it's interesting to ask for smaller values of $n$ for which $G$ is a ...
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Have you ever seen this interface? Nowadays, it is used for locking smartphones. If you haven't, here is a short video on it. The rules for creating a pattern is as follows. We must use ...
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Math puzzle: 10 digit strings generations

There was a question in a math competition that I attended last year. At the end of competition, I realized that my answer was wrong for the question below and I have never been able to figure out how ...
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Show that : $\sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2= 2 n!$

I came across this result while doing some representation theory of the permutation group $S_n$ $$\sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2 = 2 n!$$ This can be ...
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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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Calculating the number of possible paths through some squares

I'm prepping for the GRE. Would appreciate if someone could explain the right way to solve this problem. It seems simple to me but the site where I found this problem says I'm wrong but doesn't ...
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Count permutations of $\{1,2,…,7\}$ without 4 consecutive numbers - is there a smart, elegant way to do this?

Here's a problem I've solved: Count permutations of $\{1,2,...,7\}$ without 4 consecutive numbers (e.g. 1,2,3,4). So I did it kinda brute-force way - let $A_i$ be the set of permutations of $[7]$, ...
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Does the concept of permutation make sense for a set indexed by the real numbers?

I know that the concept of permutation makes sense for sequences, which are sets indexed by the natural numbers (if the sequence is infinite) or indexed by the first $n$ natural numbers (if the ...
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$A_n$ is the only subgroup of $S_n$ of index $2$.

How to prove that the only subgroup of the symmetric group $S_n$ of order $n!/2$ is $A_n$? Why isn't there other possibility? Thanks :)
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Is there a group which has precisely all finite groups as subgroups?

I would like to ask the following question: Does there exist a group $G$ such that every finite group can be embedded in $G$, and every proper subgroup of $G$ is finite? The closest ...
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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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What do all the $k$-cycles in $S_n$ generate?

Why don't $3$-cycles generate the symmetric group? was asked earlier today. The proof is essentially that $3$-cycles are even permutations, and products of even permutations are even. So: do the $3$-...
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A4 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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How to see that the polynomial $4x^2 - 3x^7$ is a permutation of the elements of $\mathbb{Z}/{11}\mathbb{Z}$

This is from Rotman's Group Theory book, although I don't have the specific reference right now, as the book is with a friend. He asks to show that $\alpha (x) = 4x^2 - 3x^7$ is a permutation of the ...
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Proof that no permutation can be expressed both as the product of an even number of transpositions and as a product of an odd number of transpositions

I am aware that there are a couple of well-known proofs of this theorem, but I'm specifically grappling with the proof given in Fraleigh's A First Course in Abstract Algebra (Theorem 9.15 in the ...
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How many possible combinations in 8 character password?

I need to calculate the possible combinations for 8 characters password. The password must contain at least one of the following: (lower case letters, upper case letters, digits, punctuations, special ...