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5 votes
0 answers
117 views

$m$-thick sets with small $n$-fold sumsets in finite cyclic groups

Problem. Is it true that for every positive integers $n,m$ there exists a subset $A_{n,m}$ of a finite cyclic group $G$ having the following two properties: $(\Sigma_n)$ the $n$-fold sum $A_{n,m}^{+...
Taras Banakh's user avatar
  • 41.9k
5 votes
0 answers
294 views

Vanishing sums of powers modulo n

Is it possible to describe all positive integer sequences $\{x_i\}$ such that $$ \sum_i x_i=n \quad \hbox{and} \quad \sum_i x_i^k\equiv 0\pmod n \quad \hbox{for all $k$ (and a given $n$)?} $$ ...
Anton Klyachko's user avatar
4 votes
1 answer
196 views

Generalization of $\lim_{n \rightarrow \infty} \prod_{i=1}^{n}\frac{2i-1}{2i}$ for a character $\chi:\mathbb{Z}/s \mathbb{Z} \rightarrow \mathbb{C}^*$

Playing with some infinite products I came up with this problem, that I'm not able to figure it out by myself. Moreover in the internet it doesn't seem to appear anywhere. Maybe it is just an easy ...
gigi's user avatar
  • 1,343
4 votes
1 answer
328 views

Maximum length of chains of subgroup in GL(n,q)

Let G=GL(n,q) be a general linear group n-dimensional over a field with q element (q power of a prime). I am looking for an estimate of maximum length of chains of subgroup in G. Thanks.
david's user avatar
  • 41
4 votes
1 answer
475 views

What natural numbers can be considered as the product of orders of elements of a finite (abelian) group

Problem A5 from putname's competition 2009 asks to prove that there is no finite abelian group such that the product of order of its elements is equal to $2^{2009}$. Starting from this problem, for ...
user avatar
4 votes
1 answer
458 views

Does there exist an order in a number field of deg>1 with a map to F_p for all p?

This question is motivated by a computational issue. Suppose $R$ is a product of orders in numberfields such that there is no ring homomorphism $R \to \mathbb Z$, then can one write an algorithm that ...
Maarten Derickx's user avatar
4 votes
0 answers
159 views

does infinite leinster groups indicates infinite perfect numbers?

The simplest definition i managed to find is: Leinster group is a finite group which her order equal the sum of its normal subgroups order, and as a perfect number is a positive integer that is equal ...
ned grekerzberg's user avatar
3 votes
1 answer
157 views

On the Upper Density of $C_2$ in finite groups

We define the upper density $\rho (G)$ of a finite group $G$ as the ratio of the number of finite groups of order $<n$ which contain a subgroup isomorphic to $G$ to the number of groups of order $&...
Evgeny T's user avatar
  • 205
3 votes
1 answer
314 views

Probability in $GL_2(\mathbb{Z}/p^{r}\mathbb{Z})$

My question may be not interesting or easy to answer ! but I am really not familiar with proba. Let $p$ be an odd prime number. and let $r\geq1$ an integer. choose an element $A\in\mathrm{GL}_2(\...
Zakariae.B's user avatar
3 votes
1 answer
608 views

Representation of GL(n, F_p) over F_p, for n small

The question is related to this post Representation theory of the general linear group over a finite prime field However, I am asking for more detailed references for n small, for example, for n=2, ...
H. Gao's user avatar
  • 31
3 votes
1 answer
288 views

Representation of a finite group over a finite field from rational representations

Suppose $G$ is a the cyclic group of $n$ elements and $p$ is a prime not in $n$. $G$ has an action on the cyclotomic field as a $\mathbb{Q}$-vector space $\mathbb{Q}[X] / \langle \Phi_n(X) \rangle$ by ...
Breakfastisready's user avatar
3 votes
0 answers
89 views

Which elements in $\mathrm{Aut}(\widehat{F_2})$ preserve the procyclic subgroup generated by the commutator $c=[a,b]$?

Let $F_2$ denote the free group over two generators $a,b$, and we denote $c=[a,b]$ as the commutator. It is well-known that any automorphism $\psi$ of $F_2$ preserves the conjugacy class of the ...
YC Su's user avatar
  • 605
3 votes
0 answers
172 views

Abelian characters and odd perfect numbers?

This question is about applications of abelian characters to odd perfect numbers: Context and Definitions: Let $n$ be a natural number and $D_n$ be the set of divisors. We can make this set to a ring ...
mathoverflowUser's user avatar
3 votes
0 answers
116 views

Ways to tell from residues modulo prime factors if $z$ is below half point

Let $N=\prod_{k=0}^{k=m}{ p_k }$ be a square-free odd integer where $p_k$ is a prime. If we are given any integer $g$ such that $0<g<N$, it is very easy to tell if $g < \frac{N}{2}$ or not. ...
ReverseFlowControl's user avatar
3 votes
0 answers
147 views

Under what conditions on $A$ and $v$ is the size of the sumset $v \cdot A + A$ over $\mathbb{F}_p$ equal or close to $|A|^2$?

Let $p$ be a prime, let $A$ be a subset of $\mathbb{F}_p$, and let $v \in \mathbb{F}_p \setminus \{0\}$. Under what conditions is $|v \cdot A + A|$ (that is, $|\{ va + b : a \in A,\ b \in A \}|$) ...
Daira-Emma Hopwood's user avatar
3 votes
0 answers
115 views

On group varieties and numbers

Suppose $\mathfrak{U}$ is a group variety. Let’s define $N_{\mathfrak{U}} \subset \mathbb{N}$ as a such set of numbers, that for any finite group $G$, $|G| \in N_{\mathfrak{U}}$ implies $G \in \...
Chain Markov's user avatar
  • 2,618
3 votes
0 answers
102 views

Localized at $p$ integral representations of finite elementary $p$-groups

Let $C_p$ be a cyclic group of prime order $p$. Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times). I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$. However, ...
Mikhail Borovoi's user avatar
2 votes
3 answers
345 views

A subgroup of index 2 for a solvable finite group transitively acting on a finite set of cardinality $2m$ where $m$ is odd

Let $G$ be a finite group acting transitively on the finite set $X=\{1,2,\dots, 2m\}$ of cardinality $2m\ge 6$ where $m$ is odd. Question 1. Is it true that $G$ always has a subgroup $H$ of index 2 ...
Mikhail Borovoi's user avatar
2 votes
1 answer
186 views

Number of homomorphism, or number of solution to equations, in finite groups

Let $G$ be a finite group, and let $P$ be a finitely generated group. Consider the number $$n=\#Hom_{Grp}(P,G).$$ It is known (see Number of solutions to equations in finite groups) that under ...
Ehud Meir's user avatar
  • 5,039
2 votes
1 answer
229 views

on the prime divisors of $(p^2+1)/2 $

The following question is equivalent to a problem in group theory. Let $ p > 13$ be a prime number distinct from 239. Let $ a=(p^2+1)/2 $. Is there any prime divisor $r$ of $a$ such that $r\mid ...
BHZ's user avatar
  • 1,168
2 votes
1 answer
193 views

Sparsity of q in groups PSL(2,q) that are K_4-simple

One of the problems that has come up during my research concerns $K_4$-simple groups (simple groups with $4$ prime divisors). The only (potentially) infinite family of groups satisfying this condition ...
abiteofdata's user avatar
2 votes
1 answer
211 views

Finite sequences which happen to be the sequence of orders of elements of a simple group

Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation $$\sum_{d|n}x_d\phi(d)=n~~~~~~~~~~~~~~~~~~~~~~~(1)$$ where $\phi$ is the Euler's totient function, $d$ ...
Minimus Heximus's user avatar
2 votes
0 answers
98 views

Estimating the cardinality of the set of conjugacy classes of subgroups in a finite group of given order

1. Let $G$ be a finite group of order $n$. I need an estimate for the number $c$ of conjugacy classes of subgroups $D\subseteq G$. Note that any subgroup of $G$ contains $1_G$, and so the set of all ...
Mikhail Borovoi's user avatar
2 votes
0 answers
235 views

Does knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ sometimes offer a significant advantage in finding $r$?

Is there a cyclic group $\mathcal G$ with generator $g$ for which the discrete log problem is assumed to be hard, but knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ for random $r$ makes finding $r$ easy? ...
wlad's user avatar
  • 4,943
1 vote
1 answer
266 views

Strange result of divisibility

I have noticed experimentally that the following question has a positive answer. Let $p>5$ and $H$ be a subgroup of $(\mathbb Z/p\mathbb Z) ^*$, with $a\in H$ and $a>2$. Is it true that $$(a-1)\...
Dattier's user avatar
  • 4,074
1 vote
3 answers
1k views

primes dividing binomial coefficients

Dear All, I am considering maximal subgroups of odd index in Alternating and Symmetric groups, and this leeds me to some questions on binomial coefficients that I presently do not know and that I ...
Ben's user avatar
  • 195
1 vote
1 answer
218 views

Average size of iterated sumset modulo $p-1$,

Given a prime $p$, what is the average size of the iterated sumset, $|kA|$, modulo $p-1$, with $p$ a prime, and $k$ given, with $A$ chosen at random? You can pick any type of prime you like for $p$, ...
Matt Groff's user avatar
1 vote
1 answer
268 views

Adding $n$-tuples over groups

Consider a finite abelian group $\mathcal{G}$. Let $S_0$ be a $n$-tuple of elements of $\mathcal{G}$, and let $S_i$ be the cyclically shifted version of $S_0$ by $i$ indices to the right. So for ...
Rahul Sarkar's user avatar
1 vote
0 answers
117 views

Reduction mod 2 for orthogonal groups

Setting Let $k$ be a real quadratic field, $\mathbb Z_k$ its ring of integers. Let $n$ be an even integer $A$ a symmetric $n$-by-$n$ matrix with coefficients in $\mathbb Z_k$. Let $L$ be the lattice $\...
Jean Raimbault's user avatar
1 vote
0 answers
94 views

A question concerning finite metacyclic groups

Consider a finite metacyclic group with presentation $$G = \langle x,y \, |\, y^n=1, x^u=y^r, x^{-1}yx = y^k \rangle,$$ where $r \mid n$. Is it true that if $G$ does not split (i.e. $G$ is a not a ...
Kashyap Rajeevsarathy's user avatar
1 vote
0 answers
127 views

Lower and upper bounds of the distance between two Frobenius numbers

I consider two sequences of numbers: $A=\{a_1,...,a_{m-1},n\}$ and $B=\{n-a_{m-1},...,n-a_1,n\}$, where $a_1 < a_2 < ... < a_{m-1} < n$ and $\gcd(A) = \gcd(B) = 1$. I investigate the ...
Виталий's user avatar
1 vote
0 answers
69 views

On equality of two quotients of a congruence subgroup

Related question: Non-torsion part of the abelianisation of congruence subgroups Let $A = \mathbb{F}_q[T]$ be the ring of polynomials with coefficients in a finite field, with $N$ a nonconstant ...
Liam Baker's user avatar
1 vote
0 answers
140 views

Does there exist an unramified $PSL_2(\mathbb{F}_p)$ extension of a quadratic field $K$?

Does there exist an unramified $PSL_2(\mathbb{F}_p)$ extension of a quadratic field $K$ for $p\geq 5$?
user avatar
1 vote
0 answers
194 views

The number of fixed points of an automorphism of $\mathbb{Z}_m\times\mathbb{Z}_n$

Let $m$ and $n$ be two positive integers such that the groups $\mathbb{Z}_m$ and $\mathbb{Z}_n$ have no common direct factor. Then an automorphism $f$ of $\mathbb{Z}_m\times\mathbb{Z}_n$ is of type $$\...
Marius Tarnauceanu's user avatar
0 votes
1 answer
283 views

Is any abelian subgroup of a semidirect product isomorphic to a direct product of abelian subgroups? [closed]

Let $H$ and $K$ be groups and $V$ an abelian subgroup of the semidirect product $\ H\rtimes K$. Do there exist abelian subgroups $H^{\prime }\leq H$ \ and $K^{\prime }\leq K$ \ such that $V\cong H^{\...
Nourddine Snanou's user avatar
0 votes
1 answer
213 views

A particular permutation on $\mathbb{Z}_n$

Let us consider $\mathbb{Z}_N=\{0,1,\cdots N-1\}$. Does there exist any permutation $\sigma$ on $\mathbb{Z}_N$ satisfying : $$\exists p,q\in\mathbb{N}, ~~p,q\neq0 (mod~N) ~~\forall i,j\in \mathbb{Z}...
ABB's user avatar
  • 4,058
0 votes
0 answers
112 views

Definition of "tame $p$-part of $\chi$"

At the beginning of Haruzo Hida's article "Big Galois representations and $p$-adic L functions", he has defined $$\chi_1= \textrm{the } N_0 \textrm{-part of} \chi \times \textrm{the tame } p\textrm{-...
tanjia's user avatar
  • 337
0 votes
1 answer
223 views

Greatest common divisor of two specified sequences of numbers (search for equality)

I consider two sequences of numbers $A=\{a_1,...,a_n\}$ and $B=\{k-a_1,...,k-a_n\}$, where $a_1 \le a_2 \le ... \le a_n \le k$. I am looking for such conditions under which: $\gcd(a_1,...,a_n) = \gcd(...
Виталий's user avatar

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