Related papers: Local commensurability graphs of solvable groups

This paper aims at studying solvable-by-finite and locally solvable maximal subgroups of an almost subnormal subgroup of the general skew linear group $\GL_n(D)$ over a division ring $D$. It turns out that in the case where $D$ is…

We introduce and investigate the solvable graph $\Gamma_\mathfrak{S}(L)$ of a finite-dimensional Lie algebra $L$ over a field $F$. The vertices are the elements outside the solvabilizer $\sol(L)$, and two vertices are adjacent whenever they…

We say A is a quasi-normal subgroup of the group G if the commensurator of A in G is all of G. We develop geometric versions of commensurators in finitely generated groups. In particular, g is an element of the commensurator of A in G iff…

Each group G of nxn permutation matrices has a corresponding permutation polytope, P(G):=conv(G) in R^{nxn}. We relate the structure of P(G) to the transitivity of G. In particular, we show that if G has t nontrivial orbits, then…

For a subgroup $H$ of a finite group $G$, the Frobenius graph $\Gamma(G, H)$ records the constituents of the restrictions to $H$ of the irreducible characters of $G$. We investigate when this graph has diameter 3.

Given a finite group $G$, the difference graph of $G$, denoted by $\mathcal{D}(G)$, is the difference of the enhanced power graph of $G$ and the power graph of $G$, with all isolated vertices removed. This paper mainly studies the…

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We attach to $N$ two graphs ${\Gamma}_G(N)$ and ${\Gamma}^{\ast}_G(N)$ related to the conjugacy classes of $G$ contained in $N$ and to the set of primes dividing the sizes…

Let $G$ be a non-abelian finite simple group. In addition, let $\Delta_G$ be the intersection graph of $G$, whose vertices are the proper nontrivial subgroups of $G$, with distinct subgroups joined by an edge if and only if they intersect…

We complete the description of semistable models for modular curves associated with maximal subgroups of $\mathrm{GL}_2 ({\mathbb F}_p )$ (for $p$ any prime, $p>5$). That is, in the new cases of non-split Cartan modular curves and…

We construct a family of finite special 2-groups which have commuting graph of increasing diameter

Let $G$ be a finite group. Denoting by ${\rm{cd}}(G)$ the set of the degrees of the irreducible complex characters of $G$, we consider the {\it character degree graph} of $G$: this is the (simple, undirected) graph whose vertices are the…

We are interested in the algebraic properties of groups of local biholomorphisms and their consequences. A natural question is whether the complexity of solvable groups is bounded by the dimension of the ambient space. In this spirit we…

We generalize ultraproducts and local-global limits of graphs to hypergraphs and other structures. We show that the local statistics of an ultraproduct of a sequence of hypergraphs are the ultralimits of the local statistics of the…

We study the spatial Gibbs random graphs introduced in [MV16] from the point of view of local convergence. These are random graphs embedded in an ambient space consisting of a line segment, defined through a probability measure that favors…

This paper studies intersections of principal blocks of a finite group with respect to different primes. We first define the block graph of a finite group $G$, whose vertices are the prime divisors of $|G|$ and there is an edge between two…

Morgan and Parker have proved that if $G$ is a group satisfying the condition that $Z(G) = 1$, then the connected components of the commuting graph of $G$ have diameter at most $10$. Parker has proved that if in addition $G$ is solvable,…

We study soluble groups G in which each subnormal subgroup H with infinite rank is commensurable with a normal subgroup, i.e. there exists a normal subgroup N such that the intersection of H and N has finite index in both H and N. We show…

Let $X$ be a compact Riemann surface of genus $g\geq 2$. Let $Aut(X)$ be its group of automorphisms and $G\subseteq Aut(X)$ a subgroup. Sharp upper bounds for $|G|$ in terms of $g$ are known if $G$ belongs to certain classes of groups, e.g.…

Among (regular, normal) parabolic geometries of type $(G,P)$, there is a locally unique maximally symmetric structure and it has symmetry dimension $\dim(G)$. The symmetry gap problem concerns the determination of the next realizable…

Let $G$ be a finite group and construct a graph $\Delta(G)$ by taking $G\setminus\{1\}$ as the vertex set of $\Delta(G)$ and by drawing an edge between two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. Let $K(G)$ be the set…