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G graph connections for finite groups G

In my research, I have seen G graph connections usually when G is a Lie group and the graph is the fatgraph of a (punctured) surface. This is usually in a physics context. However, I am curious to ...
Andrea B.'s user avatar
  • 495
2 votes
0 answers
76 views

Anti-flag transitive projective planes

Let $\Gamma$ be an axiomatic projective plane, and suppose its automorphism group acts transitively on the anti-flags (the point-line pairs $(u,V)$ such that $u$ is not incident with $V$). In the ...
THC's user avatar
  • 4,547
2 votes
0 answers
203 views

Can the relation between count of commuting pairs and conjugacy classes for finite groups be generalized to semigroups?

It is well-known that number of pairs of commuting elements in finite group G is equal to number of conjugacy classes multiplied by cardinality of G. More generally here (MO275769) Qiaochu Yuan ...
Alexander Chervov's user avatar
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 ...
Zhi-Wei Sun's user avatar
  • 15.6k
2 votes
0 answers
78 views

Width of symmetric groups

MSE crosspost For any (finite) group $G$ its length $l(G)$ is the length of maximal chain of proper subgroups (it's known and pretty widely used invariant). But we can also define width function $w_G(...
Denis T's user avatar
  • 4,600
2 votes
0 answers
84 views

A lattice ordered by inclusion and isomorphic to the lattice of quotient groups of a finite group

Let $G$ be a finite group. Consider the lattice $$L=\{ G/N:\text{$ N $ is a normal subgroup of $G $}\},$$ where $G/N \leq G/K$ if and only if $K\leq N$. The lattice operations ∧ and ∨ on quotient ...
Farid Aliniaeifard's user avatar
2 votes
0 answers
85 views

Permutation factorizations according to number of generated orbits

Let $\pi$ be a permutation in $S_n$ with cycle type $\lambda$. How many factorizations into two factors $\pi=\sigma_1\sigma_2$ are there, such that the subgroup $\langle \sigma_1,\sigma_2\rangle$ ...
Marcel's user avatar
  • 2,552
2 votes
0 answers
186 views

Sum of reciprocals in finite fields

Let $p$ be an odd prime number which large enough. I am interested in the study of the sums of reciprocals in the field $\mathbb{F}_p$. In particular, I have the following question: which primes $p$ ...
Zakariae.B's user avatar
2 votes
0 answers
140 views

About the eigenvectors of a matrix related to a Cayley graph

In some papers about the cayley graphs of finite groups the behaviour of the eigenvalues and eigenvectors of $\phi$ were discussed when $\phi=\sum_{g\in G} \lambda_G(g)$ and $\lambda_G(g)$ is defined ...
Maja's user avatar
  • 21
2 votes
0 answers
110 views

How many symmetric strings a permutation fixes

Let $A$ be an alphabet of $N$ symbols. Let $S_n$ be the group of permutations of $n$ symbols. A permutation acts on a string of letters from $A$ in the obvious way. If I ask, given a permutation $\pi\...
thedude's user avatar
  • 1,549
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 ...
Sebastien Palcoux's user avatar
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 ...
Sebastien Palcoux's user avatar
1 vote
1 answer
155 views

Max order for which connected Cayley Graphs are known to be Hamiltonian

There is a well-known conjecture that all connected Cayley graphs are Hamiltonian. For how large a value of n has the conjecture been verified (i.e., for all groups whose order is at most n)?
Geoffrey Exoo's user avatar
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 $p$-...
Nourr Mga's user avatar
  • 181
1 vote
1 answer
156 views

Sub-circle-free Christmas-gift-giving

Suppose there is a group of $n$ people giving gifts to one another. Everybody brings a gift but we want the gifts to be "well-distributed" in the group. By this I mean the following: In how many ...
Petra Schwer's user avatar
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 (...
Sebastien Palcoux's user avatar
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 ...
Raghav Kulkarni's user avatar
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. ...
Christoph Mark's user avatar
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 ...
Ben's user avatar
  • 195
1 vote
1 answer
80 views

What are the efficient algorithms to compute Hamiltonian paths on Cayley graphs of finite groups ? Can GAP do it?

The famous Lovasz conjecture predicts existence of the Hamiltonian path on Cayley graphs. In general finding such a path is NP-complete problem, but there are many heuristic algorithms. Question 1: ...
Alexander Chervov's user avatar
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 $p$, ...
Matt Groff's user avatar
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 ...
Rahul Sarkar's user avatar
1 vote
1 answer
139 views

Is an Eulerian subgroup lattice boolean?

Let $G$ be a finite group and $\mu$ the Möbius function of the subgroup lattice $\mathcal{L}(G)$. The reduced Euler characteristic of the order complex of the coset poset $\{ Kg \ | \ K<G, \ g \...
Sebastien Palcoux's user avatar
1 vote
0 answers
175 views

Random walk on N-Rubik cube group is going like sqrt(number of moves) or linear (number of moves) or? "commutative" vs. "free"(like) group pattern?

Consider higher (NxNxN) Rubik's cube group, with specific set of generators described below. What is important - that there are huge COMMUTING subsets of generators. Question: Consider a random walk ...
Alexander Chervov's user avatar
1 vote
0 answers
72 views

Scalars by which symmetrizations of cyclic permutations act on Specht modules

Let $S_n$ be the symmetric group. Pick $a \in 2,\ldots,n$ and denote by $c_a \in \mathbb{C}[S_n]$ the symmetrization of the element $(12\ldots a)$ i.e. $c_a$ is the sum of cycles of type $a$. Let $\...
Asav's user avatar
  • 163
1 vote
0 answers
144 views

Simultaneous similarity classes of pairs in $\mathrm{GL}_{n}(\Bbb Z / p\Bbb Z)$?

$\DeclareMathOperator{\GL}{\operatorname{GL}}$Let $G$ be an elementary abelian $p$-group of rank $2$. Let $\alpha, \beta :G\rightarrow \GL_{n}(\Bbb Z / p\Bbb Z)$ be two injective homomorphisms. The ...
Nourr Mga's user avatar
  • 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) = |...
Zach Hunter's user avatar
  • 3,499
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 ...
Chain Markov's user avatar
  • 2,618
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 ...
Виталий's user avatar
1 vote
0 answers
242 views

Counting elements having a given cycle structure in maximal subgroups of a generalized symmetric group

Let $G$ be the wreath product $C_7\wr S_{18}$, where $C_7$ is the cyclic group of order 7 and $S_{18}$ is the symmetric group on 18 symbols. Consider $G$ to be embedded in the group $S_{126}=S_{7\cdot ...
352506's user avatar
  • 1,021
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 ...
user112249's user avatar
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 \...
Sebastien Palcoux's user avatar
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^{\...
Nourddine Snanou's user avatar
0 votes
1 answer
130 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?
Martim Pereir's user avatar
0 votes
1 answer
155 views

Combinatorial problem in $G(54, \, 5)$ - Reprise

This post is a follow-up to my previous post MO479127. I am trying to concentrate on a subset of relations, hoping to find some structure on the set of solutions that explains why the whole set of ...
Francesco Polizzi's user avatar
0 votes
0 answers
121 views

Another question concerning finite metacyclic groups

Given a non-split finite metacyclic group $H$, does there always exist a finite split metacyclic group $G$ with a normal cyclic subgroup $N$ of prime power order such that $H \cong G/N$? Based on my ...
Kashyap Rajeevsarathy's user avatar
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(...
Виталий's user avatar
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, \...
Rodolphe's user avatar
-1 votes
1 answer
325 views

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-...
Nourr Mga's user avatar
  • 181
-1 votes
1 answer
215 views

Perfect Cayley graphs for abelian groups have $\frac{n}{\omega}$ disjoint maximal cliques

Let $G$ be a perfect/ weakly perfect Cayley graph on an abelian group with respect to a symmetric generating set. In addition let the clique number be $\omega$ which divides the order of graph $n$. ...
vidyarthi's user avatar
  • 2,089

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