# Tagged Questions

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### $PGL_d(F)$ is 2-transitive but not 3-transitive if $d > 2$

An exercise asks to prove that: If $d > 2$, then the projective general linear group $PGL_d(F)$ of dimension $d$ over a field $F$ is 2-transitive but not 3-transitive on the set of points of the ...
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### Using Plancherel's Theorem to Prove the Gauss Sum

I'm interested in proving the following: Where $p$ is an odd prime and $z$ is a primitive $p$th root of unity, we let $Q(p)=\sum^{pā1}_{k=0}z^{k^2}$. Prove: $|Q(p)|=\sqrt{p}$. Specifically, I want ...
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### Math competitions resource at university level

I want some problems especially in Algebra field for math competitions at undergraduate math students level. Does anybody here know book, website,... that I can use?!
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### Does the Cayley digraph $C$= [(12)(34),(123):$A_4$] have a Hamiltonian Circuit?

This is a problem I'm working on for a friend of mine. I haven't been able to solve it, or make much progress. I have drawn the digraph, and it consists of four directed cycles of three vertices all ...
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### Find the number of irreducible polynomials in any given degree

For any prime $p$ find the number of monic irreducible polynomials of degree $2$ over $\mathbb Z_p$. Do the same problem for degree $3$. Generalize the above statement to higher degree ...
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### Application of group theory to combinatorics

Let k be a positive integer. In how many ways one can color the edges of an equilateral triangle using k colors (two coloring schemes are considered the same if one can be obtained from the other via ...
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### Recurrence for the number of $\sigma \in S_n$ with cycle length at most $r$

I have just learned that the formula is right, but the definition of $c_n^{(r)}$ was wrong. The correct problem is: Prove $$c_{n+1}^{(r)} = \sum_{k=n-r+1}^n n^{\underline{n-k}} c_k^{(r)}$$ where ...
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### How many monoids of order three are there?

http://oeis.org/A058129 In the above link we can see the answer is 7. I have tried counting these and don't get 7. I am not sure what I am doing wrong so could someone go through counting these step ...
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### Two generating functions are combined, how are the coefficients for the combined generating function determined?

I have a function or operation $f(x)$ that takes integer inputs $0\leq x_\mathrm{in}<2^N$ to a limited set of integer outputs $x_\mathrm{out}\in[-N,N]$. I know that if all inputs are computed then ...
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### Fibonacci polynomials and factorization redux

I recently asked a question about factorizing a certain expression involving Fibonacci and Lucas polynomials. That question had a very simple and nice answer, but now I've come across another similar ...
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### A factorization problem involving Fibonacci and Lucas Polynomials

Consider a sequence of polynomial $\{w_n(x)|\, n\geq 0\}$ which are defined recursively by $w_n(x)=xw_{n-1}(x)+w_{n-2}(x)$. With $w_0(x)=0$ and $w_1(x)=1$, one gets the so-called Fibonacci polynomials ...
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### Proving that if $n\times n$ Hadamard matrix exists, then 4 divides $n$

Im looking for an explanation of the following: a standard way to prove that, if there exists Hadamard matrix of dimension $n > 2$, then $4|n$, is to suppose that without loss of generality every ...
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### 2D Rubik's cube?

There is a $3\times3$ matrix filled by numbers 1~9 that might look like this $$\begin{bmatrix}3 & 8 & 2 \\ 4 & 1 & 6 \\ 7 & 5 & 9\end{bmatrix}$$ All its rows and columns can ...
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### Intuition of Newton's identities, is it worth persuing these thoughts? Should be able to show it from special case.

I've "accidentally" stumbled on a special case of it (i=n) and it is very intuitive. Then there's some sort of "transpose" going on with the terms. So I'm thinking there might be something there. I ...
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### Understanding “avoiding sequence”

Each subsequence (91674, 91675, 91672) is called a copy, instance, or occurrence of Ļ. Since the permutation Ļ = 391867452 contains no increasing subsequence of length four, Ļ avoids 1234. ...
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### I suspect that a polynomial that is symmetric is a polynomial in “the sums of powers” - using a loose def of polynomial.

This is something I suspect, by this I mean I can't see it being wrong and I have evidence to suggest it is true. It may be a named theorem but given I've been exploring this with a notebook at a ...
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### A bijective transform that cycles. Help with definitions requested

In many ways I am a novice with mathematics. My background is college algebra. I am attempting to write my first maths paper and am faced with sifting through mathematics I am not familiar with. It ...
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EDIT (along the lines suggested by @sea turtles): Given a prime power $q$ and a positive integer $x\gt q$, how many subsets $A\subseteq{\bf F}_2^q$ have size $x$ and contain a subset $B\subseteq A$ ...
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### An algebraic proof that a set has $2^n$ subsets. (I'm looking for an inductive argument.)

There will be duplicates of this, so let me explain why I am asking: I have become blind to what it may be, so I want hints. I am blind because I can do it "combinatorially". The question wishes me ...
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### How are lopsided binomials (eg $\binom{n}{n+1})?$ defined?

For instance is $\binom{n}{n+1}=0$ always or something else?
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### Finding the coefficient of a specific term in a polynomial without expanding/simplifying?

I'm learning about Polya's Theorem which deals a lot with the coefficients of terms in a polynomial. In one problem I had the following polynomial, $(b+w)^{16}$, and the problem basically reduces to ...
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### If $|A| > \frac{|G|}{2}$ then $AA = G$

I'v found this proposition. If $G$ is a finite group , $A \subset G$ a subset and $|A| > \frac{|G|}{2}$ then $AA = G$. Why this is true ?
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### Is there a general formula for counting number of canonical forms given the minimal polynomial?

Suppose you have an operator $T$ on a vector space $F^n$, and you're given the minimal polynomial $m_T(x)=p_1(x)^{a_1}\cdots p_k(x)^{a_k}$, where $\sum a_i=\deg(m_T(x))=d\leq n$. Is there a general ...
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### How many elements of order $2$ are there in the symmetric group $S_{n}$? [closed]

How many elements of order $2$ are there in the symmetric group $S_{n}$?
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### Decomposition formulas for rotational symmetries of a cube

I have a problem that I would like to check my work on. I am also stuck on the verifications for $E$ and $F$. Any help would be greatly appreciated. Thanks in advance. Problem statement: Let $G$ be ...
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### When is round-robin scheduling possible and with in minimal time?

Suppose that you have six teams $x_0, x_1, x_2, x_3, x_4, x_5$. Can you schedule round-robin games between them so that if one game is played each day, the series of games can be completed in five ...
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### Monomial ideals: isomorphism problem for commutative algebras?

Let $I,J\unlhd K[x_1,\ldots,x_n]=K[x]$ be monomial ideals and $f\!: K[x]\to K[x]$ a graded isomorphism (given by a matrix $A=[\alpha_{i,j}]\in K^{n\times n}$, i.e. $x_i\mapsto\sum_j\alpha_{i,j}x_j$ is ...
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### How can one know how many possible abelian and non abelian group be formed from a given number of elements

Suppose that $G=\{a_1,a_2,...,a_n\}$ and how many abelian and non abelian group can be formed from this $n$ element? Attempts : I have tried consider the simple case. when n=2, there is only 1 ...
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### What does ${N\choose\mu}$ mean where $\mu = (\mu_1, … , \mu_n)$ be the vector of nonnegative integers for which $\sum_{i=1}^n \mu_i = N$?

When I've read Cohn et al. paper 2005, I've met a strange object in lemma 1.1 that looks like number of combinations: ${N\choose\mu}$ where $\mu = (\mu_1, ... , \mu_n)$ be the vector of nonnegative ...
### Maximal size of an almost-disjoint linearly independent family in $K^{\mathbb{N}}$
Let $K$ be a field, say infinite, and denote by $L$ the $K$-vector space $K^{\mathbb{N}}$. What is the maximal cardinality of a $K$-linearly independent subset $X$ of $L$ such that any two distinct ...
Let $q$ be an odd number and $g_i = (g_{i1} g_{i2} \dots g_{ir}) \in \Bbb Z_q^r$ a list of vectors with $i\in\{1,\ldots,L\}$. Let each $g_i$ have $0 < k < r$ zero entries. What is the maximum ...