All Questions
44 questions
1
vote
1
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186
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Existence of Finite Amicable Groups
I'm interested in exploring the concept of "amicable groups" as follows:
Definition. Two finite groups $G$ and $H$ are called amicable groups if:
$G$ is the direct sum of proper subgroups ...
1
vote
0
answers
189
views
The existence of solutions to linear systems of equations over the integer ring $\mathbb{Z}$
There are already detailed results on the solutions of linear equations over fields, but I'd like to inquire about any good conclusions regarding the solutions of linear equations over the integer ...
- 173
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 ...
- 14.1k
1
vote
0
answers
71
views
Bias of $a^k / q$ modulo $q$?
Let $q$ be a prime. Let $0< a < q$ be an integer so that it is primitive modulo $q$. Let $k$ be a random integer up to $q-1$. Consider
$$a^k = b_k + q * c_k$$
as $k$ varies modulo $q^2$. So $b_k$...
- 591
1
vote
1
answer
594
views
Some necessary condition for $\gcd(m,n) $ be a proper divisor of $\gcd(mk_2 +nk_1,mn) $ [closed]
Let $m,n,k_1,k_2 $ be natural numbers such that $(k_1,m)=(k_2,n)=1 $.
Statement 1: $\gcd(m,n) $ is a proper divisor of $\gcd(mk_2 +nk_1,mn) $, for every $k_1,k_2$ having the above property.
Statement ...
- 923
12
votes
1
answer
450
views
abelian quotients of permutation groups
Let $G$ be a subgroup of the permutation group $S_n$, and let $H$ be a normal subgroup of $G$ such that the quotient group $G/H$ is abelian. What is the best known upper estimate for the cardinality $...
- 1,294
16
votes
2
answers
2k
views
One question on linear combinations of roots of unity
For $n \geq 1$, I want to find all solutions $x_i$ of the equation
\begin{equation}
\begin{array}l
x_i \in \mathbb{Z}, i=0,1,2\dotsc,n-1 \\
x_i^2 = 1, i=0,1,2\dotsc,n-1 \\
\...
- 696
5
votes
1
answer
358
views
The number of polynomials on a finite group, II
This question is follow up of this MO-post.
First let us recall the necessary definitions.
A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\mathbb N$ and elements $a_0,...
- 41.8k
3
votes
2
answers
365
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Sumsets with the property "$A+B=C$ implies $A=C-B$"
Let $(G,+)$ be an abelian group and $A$, $B$ and $C$ be finite subsets of $G$ with $A+B=C$. One may conclude that $A\subset C-B$. However, $A$ need not be equal to $C-B$. What is a necessary and ...
- 429
14
votes
1
answer
1k
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A nice problem by Peter Cameron on subsets of $\{1,\dots,n\}$
Recently Professor Peter Cameron posed a number theory problem which is related to graphs of groups. The problem is:
Problem:
Let $n$ be a positive integer. Show that there exist subsets $A_1, A_2, …,...
- 4,784
2
votes
1
answer
169
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A new convolution, on function of $\mathbb F_p^n$ to $\mathbb F_p$ still zero?
Let $p$ integer prime, $f$ a function of $A=\mathbb F_p^n$ to $\mathbb F_p$, with $n\geq p+1$.
Is it true that : for all $x\in A, \sum\limits_{\sigma \in S_n} s(\sigma) \times f(x_\sigma) =0$?
$s$ ...
- 4,074
2
votes
0
answers
161
views
Monotonicity of the cycle index polynomial under restriction
The cycle index (polynomial) of the symmetric group $\mathfrak{S}_n$ is given by the formula:
$$Z(\mathfrak{S}_n)(x_1,\dots,x_n)=\sum_{1j_1+2j_2+\cdots+nj_n=n}\prod_{k=1}^n\frac{x_k^{j_k}}{k^{j_k}j_k!}...
- 43.2k
1
vote
1
answer
292
views
A conjecture on covers of $\mathbb Z$ by residue classes
Let $A=\{a_s+n_s\mathbb Z\}_{s=1}^k$ be a finite system of residue classes, where $a_s$ and $n_s>0$ are integers. For a positive integer $m$, if $A$ covers each integer at least $m$ times then we ...
- 15.6k
1
vote
1
answer
312
views
On the number of structure of $F_p[G]$-modules
Let $A$ be an abelian group and $G$ be a group. A short exact sequence of
groups like $1\longrightarrow A\longrightarrow E\longrightarrow
G\longrightarrow 1$ is called an extension. We say that $E$ is ...
- 463
1
vote
1
answer
267
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 ...
- 431
2
votes
0
answers
94
views
Is it possible to stab every permutation of any four element subset of $D_n$ with less than $n/2$ elements?
Say for a permutation group $G$ over $n$ that a set $S\subset \{1,\ldots,n\}$ is G-stabbed by $X\subset \{1,\ldots,n\}$ if for every $g\in G$ we have $gS\cap X\ne \emptyset$.
Is there for every $|S|...
- 18.7k
11
votes
3
answers
594
views
Is it possible to stab (every rotation of) any four element subset of $\mathbb Z_n$ with less than $n/2$ elements?
Say that $S\subset \mathbb Z_n$ is stabbed by $X\subset \mathbb Z_n$ if for every $t$ we have $(S+t)\cap X\ne \emptyset$.
Is there for every $|S|=4$ an $|X|<n/2$ that stabs it?
My motivation ...
- 18.7k
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 ...
- 13
8
votes
1
answer
575
views
Unstable Integers
There is a question that has been bothering my mind for quite a while now. I will present it and my current thoughts and progress on it.
Let the prime factorization of an integer $n$ be
$$n = p_1^{...
- 311
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(...
- 13
-3
votes
1
answer
961
views
Maximum element order in $S_n$ [closed]
Denote by $S_n$ the group of permutations of the set $\{1,\ldots,n\}$ with composition as binary operation. Let $m_n$ denote the maximum order that an element of $S_n$ can have. What is the smallest ...
- 48.8k
2
votes
1
answer
229
views
Has the "semidirect monoid of a semiring" been considered anywhere?
Given a semiring $S$, we get a monoid $M(S)$ as follows:
The underlying set of $S$ is $S^2$
The identity element is $(0,1)$
The law of composition is given by $$(a,A)(b,B) = (Ba+b,AB),$$ where $a,A,b$...
- 3,793
3
votes
0
answers
282
views
A new combinatorial problem for finite groups
In a recent preprint arXiv:1811.10503, I proved that if $a_1,\ldots,a_n$ are distinct elements of a torsion-free additive abelian group $G$, then there is a permutation $\pi\in S_n$ such that all ...
- 15.6k
6
votes
0
answers
226
views
A challenging problem on disjoint cosets
Motivated by my negative solutions (see this paper published in Chinese Ann. Math. 13A(1992)) to two open problems on disjoint residue calsses posed by A. P. Huhn and L. Megyesi [Discrete Math. 41(...
- 15.6k
3
votes
4
answers
654
views
A generalization of Landau's function
For a given $n > 0$ Landau's function is defined as $$g(n) := \max\{ \operatorname{lcm}(n_1, \ldots, n_k) \mid n = n_1 + \ldots + n_k \mbox{ for some $k$}\},$$
the least common multiple of all ...
- 798
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^{\...
- 463
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 $&...
- 205
2
votes
2
answers
243
views
$n$-distant permutations more than not
Let $\mathfrak{S}_{2n}$ be the permutation group of the letters $[2n]=\{1,2,\dots,2n\}$. Call a permutation $\pi\in\mathfrak{S}_{2n}$ has an $n$-distant pair if there is some $j\in [2n-1]$ such that $\...
- 43.2k
14
votes
2
answers
1k
views
$n!$ divides a product: Part I
Question. The following is always an integer. Is it not?
$$\frac{(2^n-1)(2^n-2)(2^n-4)(2^n-8)\cdots(2^n-2^{n-1})}{n!}.$$
John Shareshian has supplied a cute proof. I'm encouraged to ask:
...
- 43.2k
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(\...
- 347
5
votes
1
answer
326
views
Generalizing Kasteleyn's formula even more?
Inspired and intrigued by this question, I decided just for fun to throw in another integer into the factors and look what happens. So for $k\in\mathbb Z$, let us define $$K_r(n,k):=\prod_{\ell_1=1}^...
- 13.4k
4
votes
1
answer
178
views
finding $\mathcal{F}$ or classes in $\mathcal{F}$
Let $\mathcal{F}$ be the set of functions (sequences, if you like) $f:\mathbb{Z}_{\geq0}\rightarrow\mathbb{Z}$ such that
(a) $f(0)$ is odd, and
(b) $2^{n+1}\,\vert\,(\Delta^nf)(x)$ for all $n\geq1$ ...
- 43.2k
18
votes
3
answers
745
views
Number of primitive $n$th roots with positive versus negative real parts
Does anyone know a reference to the following results, which I can prove, but I suspect may be known. Let $R(n)$ denote the number of primitive $n$th roots of unity with positive real part, and $L(n)$ ...
- 1,991
11
votes
3
answers
1k
views
A problem on a specific integer partition
Let $n$ be a positive integer, we consider partitions of the following form :
$$n = d^{2}_{1} + d^{2}_{2} + ... + d^{2}_{r}$$ such that :
$d_{i}\vert n$
$1=d_{1}<d_{2} \le d_{3} \le ... \le d_{r}$...
3
votes
2
answers
1k
views
A problem about Determinant of sum of permutation matrices
Let $w_1$ and $w_2$ be two permutations of $\{1, \cdots , k\}$ such that for all $1\leq i \leq k$, $w_1(i)\neq w_2(i)$. Let $m$ and $n$ be two relatively prime integers. Then is there exist two ...
- 897
1
vote
1
answer
240
views
Group with 2 orbits on the nonnegative integers -- description of the orbits
Definition: Let $r(m)$ denote the residue class $r+m\mathbb{Z}$,
where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$,
let the class transposition $\tau_{r_1(m_1),r_2(m_2)}...
- 19.6k
12
votes
0
answers
558
views
Possible orders of products of 2 involutions which interchange disjoint residue classes of the integers
Definition / Question
Definition: Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where
$0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition
$...
- 19.6k
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 ...
- 195
8
votes
1
answer
417
views
Cayley Graphs of Z/nZ with invertible adjacency matrices
Let $G = \mathbb Z/n\mathbb Z$ and let $\emptyset\neq S\subseteq G$. Then the Cayley digraph of $G$ with respect to $S$ has vertex set $G$ and directed edges of the form $g\rightarrow gs$ with $s\in ...
- 38.6k
20
votes
2
answers
2k
views
Sums of powers mod p
For prime $p > 7$ with $p-1=rs$, $r>1$, $s>1$, let $A=\{x^r|x \in \mathbb{Z}_p\}$ and $B = \{x^s|x \in \mathbb{Z}_p\}$. If $g$ is a primitive root mod $p$ then $A = \{0\} \cup \{g^{ir}|0 \leq ...
- 619
7
votes
1
answer
569
views
Upper bound for size of subsets of a finite group that contains a sum-full set
Problem
I'm looking for an upper bound for the number $k(G)$ of a finite group $G$, defined as follow:
Let $\mathcal{F}_k$ be the family of subsets of $G$ with size $k$, and we
define $k(G)$ be ...
- 1,250
8
votes
1
answer
601
views
What does the expression count?
Let $q \geq 2$. What does the expression $(q^n-1)(q^n-q)(q^n-q^2)(q^n-q^3)\ldots(q^n-q^{n-1})/n!$ count? If $q$ is a prime power, then this is
the number of bases of an $n$-dimensional vector space ...
- 897
31
votes
1
answer
2k
views
Navigating $\mathbb{Z}/p\mathbb{Z}$
$\newcommand{\Z}{\mathbb{Z}}$Let's consider a silly-looking question first. Consider $\Z/p\Z$. Say I am allowed the two operations $x\mapsto x+1$ and $x\mapsto 2x$. Then, starting from $0$, I can ...
- 20.2k
8
votes
1
answer
489
views
Elements living in the conjugacy class and in the centralizer of an $m$-cycle in $A_m$
Let $m>1$ be an odd natural number, $x$ a $m$-cycle in $A_m$, the alternating group in $m$ letters, $C$ the conjugacy class of $x$ in $A_m$.
Question: How can I describe the elements in the set $\{ ...
- 81