All Questions
17 questions
8
votes
1
answer
304
views
The growth rate of a commutator set in a non-elementary group
Let $G$ be a non-elementary group generated by a finite set $S$. Here, a group is called non-elementary if it is not virtually abelian. Denote $S^{\le n}:=\{g\in G: |g|_S\le n\}$ for any $n\in \mathbb ...
- 145
11
votes
2
answers
661
views
$\mathbb Z/p\mathbb Z=A\cup(A-A)$?
$\newcommand{\Z}{\mathbb Z/p\mathbb Z}$
Can one partition a group of prime order as $A\cup(A-A)$ where $A$ is a subset of the group, $A-A$ is the set of all differences $a'-a''$ with $a',a''\in A$, ...
- 23k
2
votes
0
answers
125
views
Almost subgroups of $\mathbb S^1$
Suppose $X\subset \mathbb S^1$ is a finite subset of the group $\mathbb S^1$ such that $|X+X|<(1+c )|X|$ for a sufficiently small $c\in(0,1)$. I believe that in such case there exists a subgroup $G=...
- 14.3k
4
votes
1
answer
270
views
A combinatorial problem on abelian groups
In a 1952 paper M. Hall proved that if $G=\{a_1,\ldots,a_n\}$ is an additive abelian group of order $n$ and $b_1,\ldots,b_n$ are elements of $G$ with $b_1+\ldots+b_n=0$ then we have
$$\{a_{\sigma(i)}+...
- 15.6k
-2
votes
1
answer
353
views
What are the consequence of Snevily's conjecture to analytic number theory if really there is a connection between them? [closed]
Snevily's conjecture it is an open conjecture in Group theory for non cyclic Group and it were proved for abelian groups of prime order using a fairly standard application of the Alon-Tarsi ...
1
vote
0
answers
247
views
Sidon sets in finite groups
Suppose $G$ is a group, $S \subset G$. Let’s call $S$ a Sidon subset iff $\forall$ quadruples $(a, b, c, d)$ of distinct elements of $S$ we have $ab \neq cd$ (named after Simon Sidon who studied such ...
- 2,618
3
votes
0
answers
282
views
A new combinatorial problem for finite groups
In a recent preprint arXiv:1811.10503, I proved that if $a_1,\ldots,a_n$ are distinct elements of a torsion-free additive abelian group $G$, then there is a permutation $\pi\in S_n$ such that all ...
- 15.6k
3
votes
1
answer
153
views
On decomposition of finite Abelian groups
It is easy to see that for any finite Abelian group $G$ and any numbers $a,b$ with $|G|=ab$ there exist a subgroup $A\subset G$ and a subset $B\subset G$ such that $|A|=a$, $|B|=b$ and $G=A+B$, where $...
- 41.8k
16
votes
1
answer
455
views
Escaping from a centralizer
Let $G = Sym(n)$, $n$ even. Let $H<G$ be the stabilizer of the partition $\{\{1,2\},\{3,4\},\dotsc,\{n-1,n\}\}$, or, what is the same, the centralizer of $(1\;2) \dotsc (n-1\; n)$.
By Stirling's ...
- 20.2k
4
votes
1
answer
223
views
A permutation problem for finite subsets of an abelian group
Here I ask the following question in additive combinatorics.
QUESTION: Let $A$ be any finite subset of an additive abelian group $G$ with $|A|=n>3$. Can we write $A$ as $\{a_1,\ldots,a_n\}$ so ...
- 15.6k
3
votes
1
answer
277
views
(Extremal) arithmetic combinatorics in non-abelian groups
Roth's Theorem states that any subset $A$ of $\{1, \dots, n\}$ with no solution to the equation $$x + y = 2z,\, (x, y, z) \in A^3,\, x \neq y$$ has size $o(n)$. Similar results hold when dealing with ...
- 1,561
6
votes
1
answer
332
views
Zero-sum sets in union-closed families
The Davenport constant $D(G)$ of a finite abelian group $G$ is the minimum integer $n$ such that whenever $a_1, \ldots, a_n \in G$ (not necessarily distinct), there is a non-empty $I \subseteq [n]$ ...
- 1,169
5
votes
2
answers
387
views
Size of distinct sums in A
Let $G$ be an abelian group. Let $A\subset G$ be a finite set. $\sum_A$ is defined as: $$\left\{\sum_{b\in B}b \mid B\subset A\right\}$$ Is there any result similar to Freiman's Theorem for $\sum_A$? ...
- 901
8
votes
0
answers
304
views
A strong sum-product "for translates" in finite fields
In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting?
Proposition There exists an absolute constant $c$ such ...
- 11.2k
0
votes
0
answers
39
views
Minimum number of solutions in a system of equalities and non-equalities
Let $k<N$ and $P_1, ..., P_{2k+1}, \lambda_1, ..., \lambda_k$ be elements of a finite group $G$ of size $N$.
Find the minimum number of solution of the system
$$P_{2i} + P_{2i+1} = \lambda_i, \...
- 43
10
votes
1
answer
971
views
Is it known that $(F_p^{\times} \ltimes F_p, F_p)$ is not a relative expander family?
Shalom [edit: originally M. Burger] showed that the pair $(\mathrm{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2, \mathbb{Z}^2)$ has Relative Property (T) with respect to standard generating sets.
(The ...
- 915
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