All Questions
Tagged with finite-groups co.combinatorics
190 questions
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}:...
CommunityBot
- 1
7
votes
1
answer
399
views
Maximum size of $k$-Sidon set over $\mathbb{F}_2^n.$
Fix $k \in \mathbb{N}$, $k \ge 2.$
Does there exist a subset $A \subset \mathbb{F}_2^n$ such that $|A| \ge c 2^{n/k}$ with some absolutely positive constant $c,$ and satisfying
$$ a_1 + a_2 + \...
- 79.9k
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$, ...
CommunityBot
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23
votes
5
answers
5k
views
Has any attempt been made to classify finite groupoids?
I recently stumbled upon the Mathieu groupoid and I found them fascinating.
It appears as a subset of $S_{13}$ which is not closed under multiplication, but it turns out to be a groupoid with 13 ...
- 12.3k
1
vote
1
answer
155
views
The least common multiple of all degrees of a finite Coxeter group and indecomposable elements in the generalized cycle decomposition
This question is a follow-up of the previous question and especially the last comment therein.
Let $(W,S)$ be a finite Coxeter system with reflections $T$. Let $\ell_T$ be the reflection length. ...
- 366
4
votes
2
answers
486
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Transposition Cayley graphs are planar
Consider the Cayley graph $G$ with vertex set the elements of the symmetric group $S_n$ and generating set the set of minimal transposition generators of the group $S_n$, that is the set $S=\{(12),(13)...
5
votes
2
answers
567
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Orbits of independent sets of the hypercube
How does one enumerate the distinct orbit classes of independent sets of the hypercube modulo symmetries of the hypercubes?
The counting of the number of independent sets in an $n$-dimensional ...
- 63.9k
5
votes
2
answers
1k
views
Cardinality of certain subsets in vector spaces over finite fields
Assume that you have an $n$-dimensional vector space over a finite field (therefore the number of elements in the vector space is finite) and $F$ is a subset of this vector space which contains $m$ ...
- 63.9k
8
votes
2
answers
4k
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Order of product of group elements
Let $G$ be a finite non-commutative group of order $N$, and let $x, y \in G$. Let $a$ and $b$ be the orders of $x$ and $y$, respectively. Can we say anything non-trivial about the order of $xy$ in ...
- 63.9k
7
votes
1
answer
145
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How to prove the relationship between Stern's diatomic series and Lucas sequence $U_n(x,1)$ over the field GF(2)?
I found the bit count of Lucas sequence $U_n(x,1)$ over the field GF(2) is Stern's diatomic series, I want to know the reason?
https://oeis.org/A002487 : Stern's diatomic series
https://oeis.org/...
- 50.8k
28
votes
1
answer
2k
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Multiplying all the elements in a group
Let $G = \{ g_i | i = 1, ...,n \}$ be a finite group and denote by $G!$ the multiset consisting of all the products of all different elements of $G$ in any order, that is
$$ G! = [ \prod_i g_{\sigma(i)...
- 1,889
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 ...
- 30.1k
3
votes
1
answer
774
views
What's concrete model for Coxeter complexes?
We know for every Coxeter system $(W,S)$ there is a Coxeter complex associated by its cosets of parabolic subgroups. In Wachs's note Poset Topology p.12-13 she mentioned for the Coxeter complex of ...
- 24.2k
-1
votes
1
answer
325
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Isomorphism classes of split extensions [closed]
Let $p$ be a prime number and $n$ an integer such that $p\geq n$. Let $P(n)$ denotes the number of partitions of $n$. Can we conclude from Theorem 1.1 and Theorem 1.3 in the reference FINITE_p-...
- 181
1
vote
0
answers
80
views
Packing almost-subgroups into a group
We consider a group finite $G$. We say a set $A\subset G$ injects a set $B$ if $|A+B| = |A||B|$, and let $I(B) = \max \{|A| :A\text{ injects } B\}$.
For a subgroup $H$, it is well-known that $I(H) = |...
CommunityBot
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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
16
votes
1
answer
395
views
Geometric interpretation of the exceptional isomorphism $PSp(4,3)=PSU(4,2^2)$
It is well-known that there is an isomorphism between $PSp(4,3)$ (the symplectic group of dimension $4$ over $\mathbb F_3$) and $PSU(4,2^2)$ (the unitary group defined by $4\times4$ unitary matrices ...
- 11.2k
1
vote
1
answer
255
views
Number of commuting pairs in p-group
Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z}
)$. Consider the set $U$ of upper-triangular matrices of $G$ having
entries of $1$ on the diagonal. The set $U$ is a Sylow $p$-...
- 44.4k
12
votes
2
answers
406
views
Does asymmetric fraction of finite groups tend to $0$?
Let’s define asymmetric fraction of a finite group $G$ as the number $$\mathrm{af}(G) = \frac{|\{(g, a) \in G \times \mathrm{Aut}(G)\mid a(g) = g\}|}{|G|\cdot|\mathrm{Aut}(G)|}.$$ Equivalently it can ...
- 44.4k
7
votes
1
answer
583
views
Wreath product $S_k\wr S_n$ inside $S_{kn}$
I want to understand wreath products a little better.
Currently, my intuition about them is as follows. Take $nk$ disks, pile them in order forming $n$ piles of $k$ disks (one pile contains disks {1,...
- 9,501
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 $\{ ...
- 1,388
5
votes
1
answer
764
views
Conjugacy classes in $GL_{n}(Z / pZ)$
Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z}
)$. Consider the set $U$ of upper-triangular matrices of $G$
having entries of $1$ on the diagonal. The cardinality of $U$ is $p^{\...
- 9,501
5
votes
1
answer
275
views
Diameter of Cayley graphs of finite simple groups
Babai, Kantor and Lubotzky proved in 1989 the following theorem (Sciencedirect link to article).
THEOREM 1.1. There is a constant $C$ such that every nonabelian finite simple group $G$ has a set $S$ ...
- 63.9k
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, $|\{ va + b : a \in A,\ b \in A \}|$) ...
5
votes
0
answers
196
views
Are finite groups of exponent $d$ rare for $d \neq 4$?
Is there a way to prove, that $\lim_{n \to \infty} \frac{\text{the number of all groups of exponent }d \text{ and order less than }n}{\text{the number of all groups of order less than } n} = 0$ for $d ...
- 9,650
9
votes
1
answer
357
views
Is there a way to prove, that $2$-generated groups are rare among finite groups?
Is there a way to prove, that $\lim_{n \to \infty} \frac{\text{the number of all } 2 \text{-generated groups of order less than }n}{\text{the number of all groups of order less than } n} = 0$?
This ...
- 35.5k
3
votes
1
answer
193
views
Maximal factorization of finite simple groups and no extra intermediate
The book The maximal factorizations of the finite simple groups and their automorphism groups (by Martin W. Liebeck, Cheryl E. Praeger and Jan Saxl) provides a classification of all the triples $(G,A,...
- 44.4k
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 ...
50
votes
6
answers
11k
views
Generating finite simple groups with $2$ elements
Here is a very natural question:
Q: Is it always possible to generate a finite simple group with only $2$ elements?
In all the examples that I can think of the answer is yes.
If the answer is ...
- 63.9k
40
votes
1
answer
2k
views
Orders of products of permutations
Let $p$ be a prime, $n\gg p$ not divisible by $p$ (say, $n>2^{2^p}$). Are there two permutations $a, b$ of the set $\{1,...,n\}$ which together act transitively on $\{1,2,...,n\}$ and such that all ...
CommunityBot
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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
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(...
- 4,713
3
votes
2
answers
178
views
Asymptotics of regular semisimple elements in finite groups of lie type
Let $G$ be a reductive algebraic group defined over $\mathbb{F}_{q}$. Let $G(\mathbb{F_{q}})$ denote the finite group of fixed points under the Frobenius map $F$. Now, we know that the set of regular ...
- 14.2k
35
votes
3
answers
3k
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Does the hypergraph structure of the set of subgroups of a finite group characterize isomorphism type?
Question
Suppose there is a bijection between the underlying sets of two finite groups $G, H$, such that every subgroup of $G$ corresponds to a subgroup of $H$, and that every subgroup of $H$ ...
- 5,654
3
votes
0
answers
130
views
Graeco-Latin squares and outer-automorphisms
It is well known that $n=6$ is the only number greater than two in which there is no Graeco-Latin square of order $n$. It is also well known that $n=6$ is the only number greater than two in which ...
- 2,649
5
votes
1
answer
316
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Connected permutation groups and wreath product
Let $G$ and $H$ be subgroups of the symmetric groups $\mathfrak S_m$ and $\mathfrak S_n$. Assume that $n>1$ and that $H$ is a 'connected' permutation group, that is, there is no non-trivial $H$-...
- 37.4k
29
votes
3
answers
4k
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Roots of permutations
Consider the equation $x^2=x_0$ in the symmetric group $S_n$, where $x_0\in S_n$ is fixed. Is it true that for each integer $n\geq 0$, the maximal number of solutions (the number of square roots of $...
- 44.4k
2
votes
0
answers
116
views
On covers of groups by cosets
Suppose that ${\cal A}=\{a_sG_s\}_{s=1}^k$ is a cover of a group $G$ by (finitely many) left cosets with $a_tG_t$ irredundant (where $1\le t\le k$). Then the index $[G:G_t]$ is known to be finite. In ...
- 15.6k
3
votes
0
answers
303
views
Growth functions of finite group - computation, typical behaviour, surveys?
Looking on the growth function for Rubik's group and symmetric group, one sees rather different behaviour:
Rubik's growth in LOG scale (see MO322877):
S_n n=9 growth and nice fit by normal ...
- 4,713
6
votes
0
answers
1k
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Frobenius formula
I know two formulas by the name of Frobenius.
The first one computes the number
$$\mathcal{N}(G;C_1,\dotsc,C_k):=|\{(c_1,\dotsc,c_k)\in C_1 \times \cdots \times C_k\:|\:c_1\cdots c_k=1\}|,$$
where $...
- 63.9k
10
votes
2
answers
1k
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A cancellation property for permutations?
Let $S_n$ be the group of $n$-permutations. Denote the number of inversions of $\sigma\in S_n$ by $\ell(\sigma)$.
QUESTION. Assume $n>2$. Does this cancellation property hold true?
$$\sum_{\...
- 24.9k
13
votes
1
answer
2k
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Number of positions of Rubik's cube grows with multiplier 13 with the distance - what are explanations and groups with similar growth pattern?
Rubik's cube and its generalizations attracts certain attention of mathematical community. It is somehow "noteworthy" that it has been proved that diameter of the Rubik's cube group is 20, i.e. ...
- 52.3k
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 ...
CommunityBot
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12
votes
0
answers
191
views
Non-Boolean Eulerian interval of finite groups
An Eulerian subgroup lattice is Boolean (see here), so it is natural to wonder whether it is also true for an interval of finite groups. The smallest non-Boolean Eulerian lattice is the following:
It ...
4
votes
0
answers
266
views
Metrics on finite groups and generalizations of central limit theorems for balls volumes (à la Diaconis-Graham)
In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation groups are considered and ...
- 63.9k
8
votes
2
answers
586
views
How hard is it to compute the diameter and the growth function of a finite permutation group of small degree?
Let $G \leq {\rm S}_n$ be a finite permutation group, and let
$S = \{g_1, \dots, g_k\}$ be a generating set for $G$ which is closed
under inversion and which does not contain the identity.
The growth ...
- 11.2k
10
votes
0
answers
194
views
Permutation groups with diameter $O(n \log n)$
I suspect that many permutation puzzles can be solved in $O(n \log n)$ moves, which has led me to the following question/conjecture:
Suppose that
1. $P_i$ for $i<k=O(1)$ are permutations on an $n$ ...
- 7,545
28
votes
1
answer
1k
views
Number of irreducible representations of a finite group over a field of characteristic 0
Let $G$ be a finite group and $K$ a field with $\mathbb{Q} \subseteq K \subseteq \mathbb{C}$.
For $K=\mathbb{C}$ the number of irreducible representations of $KG$ is equal to the number of conjugacy ...
- 26.5k
14
votes
3
answers
660
views
Which partitions realise group algebras of finite groups?
Fix an algebraically closed field $K$ (maybe of characteristic zero first for simplicity, like $\mathbb{C}$).
Given a partition $p=[a_1,...,a_m]$ of an integer $n$. We can identify $p$ with the ...
CommunityBot
- 1
6
votes
0
answers
88
views
Numbers where there is a unique group with integral character table
Call a number $n$ special in case there is a unique group of order n whose character table consists of only integers. This should be equivalent to the condition that the number of conjugacy classes ...
CommunityBot
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