All Questions
Tagged with finite-groups co.combinatorics
190 questions
9
votes
1
answer
650
views
A stronger version of a problem of Kenneth Brown using representations
Let $G$ be a finite group and $\mathcal{L}(G)$ its subgroup lattice. Let $\mu$ be the Möbius function on $\mathcal{L}(G)$.
The reduced Euler characteristic of the order complex of the coset poset $\{ ...
16
votes
1
answer
1k
views
Tensor power of the natural representation of Sn
The symmetric group $S_n$ acts over $V=\mathbb{R}^n$ by permuting the canonical basis.
So it acts over $V^{\otimes p}$ with a diagonal action (acts the same over each element of the tensor product).
...
- 583
1
vote
0
answers
285
views
Classification of transitive subgroups of finite symmetric groups generated by double transpositions
I want to classify (up to isomorphism) all transitive subgroups of symmetric group $S_n$ which are generated by double transpositions (product of two transpositions). Is there a characterization for ...
- 11
3
votes
0
answers
79
views
Groups with maximal density of the set of orders of elements
Based on the answer to this question, I am wondering:
Let $n>8$ and $G$ a finite group for which all orders of its elements are contained in $\{1,\dotsc,n\}$ (denote the set of all such groups by $\...
- 13.4k
20
votes
1
answer
586
views
$q$-(and other)-analogs for counting index-$n$ subgroups in terms of Homs to $S_n$?
The following formula of astonishing beauty and power (imho):
$$ \sum_{n \ge 0} \frac{| \mathrm{Hom}(G,S_n) | }{n! } z^n = \exp\left( \sum_{n \ge 1} \frac{|\text{Index}~n~\text{subgroups of}~ G|}nz^...
- 24.9k
13
votes
2
answers
1k
views
Number of commuting pairs (triples, n-tuples) in GL_n(F_q) (and other groups)?
Question 1 What is the number of pairs of commuting elements in GL_n(F_q) ?
I am aware of many results concerning commuting elements in Mat_n(F_q), but I am interested in GL i.e. non-degenerate ...
- 24.9k
3
votes
2
answers
293
views
An optimization problem in finite groups
Let $G$ be a finite group, say, of order $n$. I ran into the following problem (where $|A|$ denotes the cardinality of a set $A$):
To determine the minimum of $|A|+|B|$ for sets $A, B \subset G$ such ...
8
votes
0
answers
88
views
Is recognizing if a Latin square is isotopic to its transpose more efficient than computing its symmetry group?
Ihrig and Ihrig (2007) described a mathematical method for determining if a Latin square is isotopic to its transpose (where isotopic Latin squares vary by permuting the rows, columns and symbols). ...
- 1,327
13
votes
1
answer
651
views
The Möbius number of the nonabelian finite simple groups
Let $L$ be a finite lattice with minimum $\hat{0}$ and maximum $\hat{1}$. The Möbius function $\mu$ for $L$ is defined recursively by: for $\forall a,b \in L$ with $a<b$, $\mu(b,b) = 1$ and $\mu(...
8
votes
2
answers
617
views
sum-sets in a finite field
Let $\mathbb{F}_p$ be a finite field, $A=\{a_1,\dots,a_k\}\subset\mathbb{F}_p^*$ a $k$-element set, for $k<p$. $\mathfrak{S}_k=$permutation gp.
Question. Is it true there is always a $\pi\in\...
- 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
16
votes
1
answer
804
views
Existence of a faithful irreducible representation using Möbius function
Let $G$ be a finite group, $L(G)$ its subgroup lattice and $\mu$ the Möbius function.
Consider the Euler totient of $G$ defined as follows:
$$ \varphi(G) = \sum_{H \le G}\mu(H,G) |H| $$
Let $X=\{M_1, \...
9
votes
2
answers
762
views
Solutions of $x^d=1$ in the symmetric group
L Moser and M Wyman, On solutions of $x^d = 1$ in symmetric groups, Canad. J. Math., 7 (1955), pages 159-168, explored asymptotic behavior of the cardinality of such permutations:
$$f_d(n):=\#\{\pi\in\...
- 43.2k
15
votes
1
answer
679
views
Submodules of $({\mathbb Z}/6{\mathbb Z})^n$ intersecting $\{0,1\}^n$ trivially
$\newcommand{\F}{{\mathbb F}}$
$\newcommand{\Z}{{\mathbb Z}}$
Suppose that $\F$ is a finite field of prime order $p:=|\F|$, and let $n$ be a positive integer. I consider the regime where $\F$ is ...
- 23k
14
votes
0
answers
378
views
A hard Lefschetz theorem for nilCoxeter algebras
Let $W$ be a finite Coxeter group and $\mathcal{N}(W)$ its nilCoxeter
algebra (over the reals, say), as defined at
https://en.wikipedia.org/wiki/Nil-Coxeter_algebra. $\mathcal{N}(W)$ has
a natural ...
- 50.8k
5
votes
0
answers
241
views
Counting the number of orbits finite groups of "diagonal type"
Let $n$, $k$, $r_1, \dots, r_k$ be positive integers.
For each $i \in [k]:=\{1,\dots,k\}$, suppose we are given $n$ permutations of the the set $[r_i]$, that is $f_1^{(i)}, \dots, f_n^{(i)}$ in $\...
- 2,479
5
votes
1
answer
503
views
does this set of permutations form a group? And more
Consider the group of $mn\times mn$ permutation matrices $\mathfrak{S}_{mn}$ and partition each such matrix $P$ into $n^2$ blocks of $m\times m$ matrices $Q_{i,j}$. Now, transpose each $Q_{i,j}$ (...
- 43.2k
3
votes
0
answers
226
views
$S_n$ action on the sequences of transpositions
It is well-known, that any element $\rho$ of the symmetric group $S_n$ with $n-p$ cycles admits a unique presentation as a product of a sequence of transpositions $\{(a_i\,b_i)\}_{i = 1}^p$ with $a_i &...
- 401
13
votes
2
answers
1k
views
The exceptional isomorphism between PGL(3,2) and PSL(2,7): geometric origin?
It is well-known there is an isomorphism between $GL(3,2)=PGL(3,2)$, the automorphism group of the Fano plane (i.e. the projective plane over the finite field with two elements), and $PSL(2,7)$, which ...
- 35.5k
8
votes
2
answers
576
views
Two statistics on the permutation group
Let $\mathfrak{S}_n$ be the permutation group on an $n$-element set. For each fixed $k\in\mathbb{N}$, consider the two sets
$$A_n(k)=\{\sigma\in\mathfrak{S}_n\vert\,\, \text{$\exists i,\,\, 1\leq i\...
- 43.2k
3
votes
1
answer
281
views
Does the graded face poset of a BN-pair admit a EL-labeling?
Let $G$ be a finite group with a BN-pair of rank $n$. Let $B$ be the associated Borel subgroup.
Let $P$ be the poset of proper right cosets (i.e. $Kg$ with $K \in [B,G)$ and $g \in G$).
Let $\hat{P}:...
3
votes
4
answers
610
views
Factorization in the group algebra of symmetric groups
Let $S_n$ be the symmetric group on $\{1, \ldots, n\}$. Let
\begin{align}
T=\sum_{g\in S_n} g.
\end{align}
Are there some references about the factorization of $T$?
In the case of $n=3$, we have
\...
- 6,201
3
votes
1
answer
173
views
Is there an atom K of [H,G]≃B2 with |K:H|≡|G:H|(mod 2)?
Let $[H,G]$ be a rank $2$ boolean interval of finite groups.
Statement 1: There is an atom $K$ of $[H,G]$ such that $|K:H|≡|G:H|($mod $ 2)$.
The following picture illustrates the statement.
...
2
votes
0
answers
97
views
Is the bounded coset poset of a boolean interval of finite groups, Cohen-Macaulay?
Let $[H,G]$ be a boolean interval of finite groups and let $\hat{C}(H,G)$ be its bounded coset poset (i.e. the poset of cosets $Kg$ with $K \in [H,G]$, bounded below by $\emptyset$ and bounded above ...
8
votes
0
answers
435
views
A relation between intersection and product on Boolean interval of finite groups
Let $[H,G]$ be a Boolean interval of finite groups (i.e. the lattice of intermediate subgroups $H \subseteq K \subseteq G$, is Boolean). For any element $K \in [H,G]$, let $K^{\complement}$ be its ...
6
votes
1
answer
341
views
Sum of Young symmetrisers of a given shape
Preliminaries and notation:
Let $n\in \mathbb{Z}_{>0}$ and $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_s)\vdash n$ be a partition. Given a Young diagram of shape $\lambda$, we can associate it ...
- 87
7
votes
1
answer
313
views
Subgroup ranks of the symmetric group
It's well known that every subgroup $G$ of $S_n$ has a generating set of size at most $n-1$ and that this generating set can be found algorithmically (by Jerrum's filter)
I have heard many times a ...
- 539
2
votes
0
answers
154
views
Nonvanishing of the dual Euler totient on boolean intervals of finite groups
The rank $n$ boolean lattice $B_n$, is the subset lattice of $\{1,2, \dotsm n \}$.
Let $[H,G]$ be a boolean interval of finite groups. Its Euler totient is defined by $$\varphi(H,G):=\sum_{K \in ...
3
votes
1
answer
184
views
Is there a characterization of CI-groups of order less than 100?
We know some benefit criterion in articles such as:
C. H. Li, On isomorphisms of finite Cayley graphs-a survey, Discrete Math., 256 (2002) 301-334.
C. H. Li, Z. P. Lu, P. P....
- 79
1
vote
0
answers
81
views
An optimal lower bound related to generators in a boolean interval of finite groups
Let $[H,G]$ be a rank $n$ boolean interval of finite groups (i.e. $[H,G] \simeq B_n$ as lattice).
Let the set $E = \{ g \in G \ | \ \langle H,g \rangle = G \}$
Remark: If $g \in E$ then $Hg \...
2
votes
1
answer
137
views
Intransitive finite irreducible linear groups whose orbits are all large
I am interested in intransitive irreducible linear subgroups $G\subseteq\mathrm{GL}_n(\mathbb{F}_p)$ acting on $V-\{0\}=\mathbb{F}_p^n-\{0\}$ in the natural way, such that all of the orbits are very ...
- 537
5
votes
1
answer
204
views
A decomposition of $w_0$ which is similar to the reduced decomposition
Some basic definitions about reduced decomposition:
In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by adjacent ...
- 1,997
5
votes
2
answers
330
views
sum of squares of Schur polynomials indexed over partition valued functions on a set
Fix a finite set $X$ and two natural numbers $d$ and $n$.
For a partition $\lambda$ and a number $d$ denote by $s_\lambda^d(x_1,\dots,x_d)$ the Schur polynomial in $d$-many variables $x_1,\dots,x_d$. ...
- 725
6
votes
1
answer
629
views
Positivity of the alternating sum of indices for boolean interval of finite groups
Let $G$ be a finite group and $H$ a subgroup such that the interval $[H,G]$ is a boolean lattice.
Let $L_1, \dots , L_n$ be the maximal subgroups of $G$ containing $H$.
Let the alternative sum ...
1
vote
1
answer
116
views
Can the reversed lattice of a subgroups interval be represented?
Let $G$ be a finite group and $H$ a subgroup. The interval $[H,G]$ is the lattice of overgroups of $H$.
It is an open problem to know if every finite lattice can be represented by such an interval (...
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
12
votes
1
answer
682
views
A sum over characters of the symmetric group
Let $C_\mu$ be the size of the conjugacy class in $S_n$ of permutations whose cycletype is the partition $\mu\vdash n$. Let $\chi$ be the characters of the irreducible representations of $S_n$.
Let $\...
- 1,549
6
votes
0
answers
240
views
Factorization of permutations into two factors with fixed number of cycles, plus a placement condition
In my recent work I have been led to consider the following type of permutation factorizations.
Let $\pi_1$ and $\pi_2$ be two fixed permutations, with disjoint support, i.e. $\pi_1$ acts on $\{1,......
- 2,552
2
votes
3
answers
888
views
The number of submodules of $\mathbb{Z}_q^n$
Observe $\mathbb{Z}_q^n = \mathbb{Z}_q \times \cdots \times\mathbb{Z}_q$ as a module over $\mathbb{Z}_q\equiv\mathbb{Z}/q\mathbb{Z}$, for general $q$.
I am interested in the following questions:
How ...
- 503
9
votes
0
answers
297
views
An abstract zero-sum problem
I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ...
- 1,169
3
votes
0
answers
186
views
Which Dihedral Groups are $\text{CI}$-Groups?
Let $D_{n}$ denotes the dihedral group of order $2n$. Firstly, for self-referencing of the question, I give some definitions which are standard.
Let $G$ be a finite group. A subset $S$ of group $G$ ...
- 4,784
12
votes
2
answers
660
views
On shifted symmetric power sums
The functions $p^*_k(x)=\sum_{i=1}^N ((x_i-i)^k-(-i)^k)$ are analogues of power sum symmetric functions, called shifted symmetric by Okounkov and Olshanski. Define $p^*_{(k_1,k_2,...)}=p^*_{k_1}p^*_{...
- 2,552
4
votes
1
answer
491
views
What is the permutation group generated by those three given morphisms of the affine space $\mathbb{F}_q^3$?
Let $\mathbb{A}^3 = \mathbb{F}_q^3$. Consider the following three functions $\mathbb{A}^3\to\mathbb{A}^3$:
\begin{eqnarray*}
h: (x, y, z) &\mapsto& (x, y, xy - z) \\
u: (x, y, z) &\mapsto&...
user4324
1
vote
1
answer
178
views
Does every connected vertex transitive graph on $n$ vertices (except for $C_n$) have minimum feedback vertex set of size $\Omega(n)$?
Feedback vertex set is a set of vertices whose removal leaves an acyclic graph.
It is known that every vertex transitive graph on $n$ vertices has minimum vertex cover of size $\Omega(n)$. It is also ...
- 197
3
votes
0
answers
215
views
scalar multiple of Young symmetrizer
The following is a lemma from Fulton and Harris' book -Representation theory,a first course (page 53):
Lemma: For all $x\in \mathbb{C}\mathfrak{S}_r$, $c_{\lambda}\cdot x\cdot c_{\lambda}= scalar \;...
- 181
4
votes
0
answers
95
views
Is the size of maximum matching in vertex transitive 3-uniform hyper-graph on $n$ vertices always $\Omega(n)$?
What is the best known lower bound on the size of the maximum matching in a vertex transitive $3$-uniform hyper-graph?
- 197
14
votes
1
answer
865
views
Enumeration of $0-1$ matrices with determinant $1$
Has the number $f(n)$ of $n \times n$, $0{-}1$ matrices whose determinant is $+1$
been enumerated?
E.g., for $n{=}2$, there are $f(2)=3$ such matrices:
$$
\left(
\begin{array}{cc}
1 & 0 \\
0 &...
- 151k
9
votes
2
answers
403
views
Certain signed sum over $S_n$
The following question appeared in my research:
Let $G_1,G_2,G_3$ all be subgroups of $S_n$, and consider the sum
$$
\sum_{g_i \in G_i, g_1g_2g_3 = id} \epsilon(g_1)
$$
that is, we only consider ...
- 15.8k
7
votes
1
answer
146
views
Covering a set with images of a transversal
Let $G$ be a permutation group on a finite set $\Omega$ with orbits $\Omega_1,\ldots,\Omega_k$.
By a transversal we mean a set $\lbrace\omega_1,\ldots,\omega_k\rbrace$ with $\omega_j\in\Omega_j$ for ...
- 37.7k
17
votes
0
answers
824
views
What's the big deal about $M_{13}$?
$M_{13}$ is the Mathieu groupoid defined by Conway in
Conway, J. H. $M_{13}$. Surveys in combinatorics, 1997 (London), 1–11,
London Math. Soc. Lecture Note Ser., 241, Cambridge Univ. Press, ...
- 11.2k