All Questions
Tagged with gr.group-theory co.combinatorics
436
questions
4
votes
1
answer
152
views
Nonempty intersection of cosets of finite-index subgroups
$\DeclareMathOperator\lcm{lcm}$This question is crossposted from MSE.
Let $H_1,\dots,H_{n+2}$ be cosets of finite-index subgroups of $\mathbb{Z}^n$ and suppose for all $i=1,\dots,n+2$, $\bigcap_{j\neq ...
- 7,836
1
vote
0
answers
67
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 $b_k$...
- 581
5
votes
0
answers
186
views
Subgroups of the symmetric group and binary relations
Motivation
The following came up in my work recently. (NB this is the motivation, not the question I'm asking. You can skip to the actual question below, which is self-contained, but not self-...
- 333
1
vote
1
answer
295
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 ...
- 913
0
votes
0
answers
110
views
Multivariate polynomial representations of the infinite dihedral group
The presentation given in Wikipedia for the infinite dihedral group is
$$\langle r,s\mid s^2 =1, srs = r^{-1}\rangle.$$
Let $[R]$ denote the infinite set of reciprocal partition polynomials $R_n(u_1,...
- 9,407
-2
votes
2
answers
208
views
Must an isomorphism preserving graph transformation preserve the order of the automorphism group?
Let $F$ be some function graph to graph which preserve graph isomorphism.
Example of such $F$ are the line graph, the $k$-subdivision of $G$
and many others.
$F$ need not preserve the order, the ...
- 24.1k
3
votes
0
answers
103
views
Twisted permutations
We consider a set $E$ with an involution (having perhaps fixed points).
We denote orbits by $\lbrace x,\overline{x}\rbrace$ (with $\overline{x}=x$ in
the case of a fixed point).
We consider sequences $...
- 17k
2
votes
0
answers
151
views
The canonical automorphism of the symmetric group
Let $S_n$ be the symmetric group of order $n$. Denoting simple transpositions by $\sigma_i$ the collection $\sigma_1, \dots, \sigma_{n-1}$ generates $S_n$ subject to the following relations:
$$
\sigma ...
- 1,104
12
votes
1
answer
418
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 $...
- 1,046
1
vote
0
answers
61
views
Convolutions of (m)-associahedra and (m)-noncrossing partition polynomials--combinatorial proofs?
I'm looking for combinatorial proofs of the convolutional identity COP below and its specializations I) and II).
(Edit 6/2/2023: A combinatorial proof is sketched in a blog post by Mike Spivey of a ...
- 9,407
2
votes
2
answers
175
views
Decompose complete directed graph with n vertices into n edge-disjoint cycles with length n−1
I want to know how to decompose a complete directed graph with $n$ nodes into $n$ edge-disjoint cycles with length $n-1$. I found this result was proved in Bermond and Faber - Decomposition of the ...
- 21
8
votes
1
answer
301
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
14
votes
0
answers
317
views
Poset defined on pairs of subgroups
Let $G$ be a group. Consider the set $P(G)$ of all pairs $(H,N)$ of subgroups of $G$ such that $N$ is a normal subgroup of $H$. Consider the relation $\leq_G$ on $P(G)$ defined as follows: $(H,N)\...
- 1,320
6
votes
1
answer
289
views
Does the Shalen-Wagreich lemma holds for non-symmetric generating sets?
Let $G$ be a group and $H$ a subgroup with finite index $d$. Then for any finite generating set $S$ of $G$, does $S^{\le k}$ contain a generating set of $H$ where $k$ is a constant depending only on $...
- 145
4
votes
0
answers
97
views
Doubly stochastic matrices that remain doubly stochastic after conjugating by the character table of a finite abelian group
I am curious if anything is known about the following.
Let $\Gamma$ be a finite abelian group, and let $\chi$ be its character table, normalized so that it is a unitary matrix. E.g., if $\Gamma$ is $\...
- 2,084
3
votes
0
answers
206
views
A bridge between the algebraic / differential geometry of $\frak{sl}_2(\mathbb{C})$ and the Sheffer-Appell calculus and combinatorics
In "Four examples of Beilinson-Bernstein localization", Anna Romanov introduces the basis
$m_k = \frac{(-1)^k}{k!} \partial^k \delta $
on p. 9, where $\partial$ is a partial derivative and $\...
- 9,407
0
votes
0
answers
82
views
Arithmetic triangles and unimodality of its rows
Let's consider the sequence of coefficients of $\prod_{i}\frac {1-x^{d_i}} {1-x}$, where $d_i$ is a monotonically increasing nonnegative integer sequence.
How to prove that the coefficients form an ...
5
votes
2
answers
563
views
The relation $x \sim g x g$ in groups
While thinking about item (2) in Standard or good names for relations between maps, I thought I'd look at the relation $x \sim g x g$ in groups.
Going through all small groups of order at most 64, it ...
- 5,473
14
votes
2
answers
1k
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 \\
\...
- 264
11
votes
2
answers
652
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$, ...
- 22.6k
4
votes
1
answer
454
views
Turán's theorem for cosets of groups
Let $G$ be a finite group, $G',H$ be its subgroups and $H'=G'\cap H$. For each $g\in G$, we create a map $f_g:G'/H'\rightarrow G/H: aH'\rightarrow gaH$. It's easy to see that the map is well defined ...
- 1,320
8
votes
2
answers
255
views
One element commutation classes of reduced decompositions of the longest element of the Weyl group
For the symmetric group on $n$ objects $S_n$ the question of how to write its longest element $w_0$ as a reduced decomposition is an important combinatorical problem. As example, in this question the ...
1
vote
0
answers
99
views
Generalized words [closed]
Dan Segal, in his book 'Words', has defined generalized words. I have trouble understanding generalized words. What I have understood from the definition of generalized words are as follows:
Let $X = \...
- 273
1
vote
0
answers
270
views
Functional equation $f(x*y) = f(f(x)*f(y))$
Find all endo-functions $f$ on a commutative semigroup $(\mathbb{S},*)$ such that
$f(x*y) = f(f(x)*f(y))$.
Typical case of interest are $(\mathbb{N},+)$ or $(\mathbb{Z}/k\mathbb{Z},+)$ or $(\mathbb{Z}/...
- 1,286
5
votes
1
answer
354
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,...
- 40.2k
6
votes
0
answers
188
views
The highest degree of a polynomial on a finite group
This question is motivated by the comments and the answer to this MO-question.
First let us recall some definitions.
A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\...
- 40.2k
25
votes
2
answers
1k
views
The number of polynomials on a finite group
A function $f:X\to X$ on a group $X$ is called a polynomial if there exist $n\in\mathbb N=\{1,2,3,\dots\}$ and elements $a_0,a_1,\dots,a_n\in X$ such that $f(x)=a_0xa_1x\cdots xa_n$ for all $x\in X$. ...
- 40.2k
4
votes
1
answer
196
views
Representation of $\mathrm{AGL}(V)$ on the homology of the poset of affine subspaces of $V$
Let $V$ be an $n$-dimensional vector space over a finite field $F$ (of order $q$). Denote by $\mathrm{AGL}(V)$ the group of invertible affine transformations of $V$; so $\mathrm{AGL}(V)$ consists of ...
- 36.8k
5
votes
0
answers
189
views
Groups of non-orientable genus 1 and 2
The non-orientable genus (aka crosscap-number) $\overline{\gamma}(G)$ of a finite group $G$ is the minimum non-orientable genus among all its connected Cayley graphs (and $0$ if $G$ has a planar ...
- 150
7
votes
1
answer
148
views
Density of “diagonal sets” in amenable groups
Let $G$ be a countable amenable group with a (left) Følner sequence $(F_n)$. Let $\Gamma$ be a subset of $G$ that has density $1$ with respect to $(F_n)$, in the sense that
$$
\lim_{n \to \infty} \...
- 323
2
votes
0
answers
133
views
Cardinality of set of powers and roots of an element in a group $G$
For a finite group $G$ and $a \in G$, we define
$${\rm Pow}(a) := \left\{b \in G : b \in <a> \text{ or } a \in <b> \right\}.$$
is it possible to explicitly count the cardinality of this ...
- 1,209
0
votes
1
answer
288
views
Lower bound of the largest irreducible character degree of alternating group $A_n$
$\newcommand\cd{\mathrm{cd}}$Let $A_m$ and $A_n$ be two alternating groups and $15\le m+2 \le n$. Denote $\cd_m$ and $\cd_n$ as the largest irreducible character degree of $A_m$ and $A_n$, ...
- 1
3
votes
2
answers
333
views
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 ...
- 421
13
votes
1
answer
954
views
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, …,...
- 4,738
2
votes
0
answers
113
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=...
- 13.9k
1
vote
1
answer
173
views
Words representations of elements of a symmetric group
Let $S=\{(1,2),(1,2,\ldots,n),(1,n,,n-1,\ldots,2)\}$ be a subset of the symmetric group $S_n$. Let $a=(1,2),b=(1,2,\ldots,n),c=(1,n,n-1\ldots,2)$ be the elements of $S$. My question is, since $S$ is a ...
- 1,841
1
vote
0
answers
97
views
Are there standard short notations for ascending and descending cyclic permutations?
In a paper I am currently writing I use cyclic permutations of the form
$$
(k,k+1,\dots,\ell)
$$
and
$$
(\ell,\ell-1,\dots,k)
$$
of consecutive elements quite a lot (I added the commas to avoid ...
- 6,643
16
votes
0
answers
324
views
Row of the character table of symmetric group with most negative entries
The row of the character table of $S_n$ corresponding to the trivial representation has all entries positive, and by orthogonality clearly it is the only one like this.
Is it true that for $n\gg 0$, ...
- 21.7k
11
votes
1
answer
281
views
Infinite vertex-transitive graph where every automorphism has a fixed vertex
This is a follow-up to the question Connected vertex-transitive graph with the fixed-point property. In particular, it is based on a comment by user bof.
Let $G = (V,E)$ be a graph with $V$ infinite. ...
- 21.7k
14
votes
2
answers
761
views
Groupoid cardinality of the class of abelian p-groups
$\DeclareMathOperator\Aut{Aut}\newcommand\card[1]{\lvert#1\rvert}$So, after going over the classification of finite abelian groups in a class I was teaching this winter, I got curious about whether it ...
- 39.4k
5
votes
0
answers
155
views
When is a Schreier coset graph vertex transitive
When is a Schreier Coset graph on a group $G$ with subgroup $H$ and symmetric generating set $S$(without identity) vertex transitive?
It is well known that when $H$ is normal, the Schreier coset graph ...
- 1,841
158
votes
36
answers
14k
views
Conceptual reason why the sign of a permutation is well-defined?
Teaching group theory this semester, I found myself laboring through a proof that the sign of a permutation is a well-defined homomorphism $\operatorname{sgn} : \Sigma_n \to \Sigma_2$. An insightful ...
- 59k
2
votes
1
answer
166
views
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 $x\in A, \sum\limits_{\sigma \in S_n} s(\sigma) \times f(x_\sigma) =0$?
$s$ ...
- 3,609
3
votes
0
answers
99
views
Commuting probabilities for a conjugacy class of a finite simple group $G$ which generates $G$
Let $G$ be a nonabelian finite simple group. Let $C\subset G$ be a conjugacy class which generates $G$. Let $E\subset C\times C$ be the subset consisting of pairs $(c,d)$ with $cd =dc$. Define the ...
- 6,567
17
votes
1
answer
1k
views
Can this probability be obtained by a combinatorial/symmetry argument?
Suppose that $a_1,\dots,a_n,b_1,\dots,b_n$ are iid random variables each with a symmetric non-atomic distribution.
Let $p$ denote the probability that there is some real $t$ such that $t a_i \ge b_i$ ...
- 111k
4
votes
1
answer
202
views
Integer-valued polynomials from Pólya counting
Let finite group $G$ act on a finite set $X$ and hence on colorings $Y^X$, where $Y=\{1,2,\ldots,k\}$ is a set of colors. The Burnside-Pólya-Redfield-etc. counting theorem says that the number of ...
- 21.7k
2
votes
0
answers
132
views
Update on Viskov's paper on random processes, Lagrange inversion, and the Heisenberg–Weyl algebra
"A Random Walk with a Skip-Free Component and the Lagrange Inversion Formula" by Viskov presents connections among Lagrange inversion and measures of random Lévy processes. The freely ...
- 9,407
0
votes
1
answer
128
views
Number of reduced decompositions of the dihedral group $D_6$ [closed]
The Weyl group of $\frak{g}_2$ is the dihedral $D_6$. Let us denote its longest element by $w_0$. How many reduced decompositions does $w_0$ have?
- 327
6
votes
2
answers
311
views
Is there a combinatorial interpretation of this array in terms of $S_{2n+1}$?
I have recently encountered a triangular array $(a_{i,j})_{0\le i\le j}$, each line of which might (should?) have a combinatorial interpretation in terms of $S_{2n+1}$. Here it is (the first entry of ...
- 13.1k
54
votes
4
answers
5k
views
How many square roots can a non-identity element in a group have?
Let $G$ be a finite group. Let $r_2\colon G \to \mathbb{N}$ be the square-root counting function, assigning to each $g\in G$ the number of $x\in G$ with $x^2=g$. Perhaps surprisingly, $r_2$ does not ...
- 785