Related papers: Bound on the diameter of split metacyclic groups

For a finite group $G$ we investigate the difference between the maximum size MaxDim$(G)$ of an "independent" family of maximal subgroups of $G$ and maximum size $m(G)$ of an irredundant sequence of generators of $G$. We prove that…

A matrix is said to be {\it cyclic} if its characteristic polynomial is equal to its minimal polynomial. Cyclic matrices play an important role in some algorithms for matrix group computation, such as the Cyclic Meataxe developed by P. M.…

We give a lower bound for the degree of a finite cover of a hyperbolic 3-manifold which fibers over the circle, in terms of volume, the diameter of the manifold and other new invariants.

New upper bounds on the size of the torsion group of a $\mathbb{Q}$-acyclic simplicial complex are introduced which depend only on the vertex degree sequence of the complex and its dimension.

Let $d(G)$ be the minimum number of elements required to generated a group $G.$ For a group $G $ of order $p^n$ with derived subgroup of order $ p^k $ and $d(G) = d,$ we knew the order of the Schur multiplier of $G$ is bounded by $…

Let $G$ be a countable group that splits as a free product of groups of the form $G=G_1\ast\dots\ast G_k\ast F_N$, where $F_N$ is a finitely generated free group. We identify the closure of the outer space…

By using the structure and some properties of extraspecial and generalized/almost extraspecial $p$-groups, we explicitly determine the number of elements of specific orders in such groups. As a consequence, one may find the number of cyclic…

We determine upper bounds for the maximum order of an element of a finite almost simple group with socle T in terms of the minimum index m(T) of a maximal subgroup of T: for T not an alternating group we prove that, with finitely many…

Consider the Cayley graph of the cyclic group of prime order q with k uniformly chosen generators. For fixed k, we prove that the diameter of said graph is asymptotically (in q) of order q^(1/k). The same also holds when the generating set…

In this paper, we continue the study of the generator graph of a group. In 2023, Tacbobo [9] defined the generator graph of a nontrivial group to be the graph whose vertices are the elements of the group, with two vertices being adjacent if…

Given a weighted and complete graph G = (V, E), V denotes the set of n objects to be clustered, and the weight d(u, v) associated with an edge (u, v) belonging to E denotes the dissimilarity between objects u and v. The diameter of a…

We give an upper bound for the dimension of a germ of a totally geodesic submanifold, and hence of a Shimura variety of A_{g-1}, contained in the Prym locus. First we give such a bound for a germ passing through a Prym variety of a k-gonal…

Let us denote elements of the symmetric group $S_n$ using square brackets for the one-line notation. Cycles will be represented using parentheses, following the standard cycle notation. Under this convention, the full reversal of the…

The conditional diameter of a connected graph $\Gamma=(V,E)$ is defined as follows: given a property ${\cal P}$ of a pair $(\Gamma_1, \Gamma_2)$ of subgraphs of $\Gamma$, the so-called \emph{conditional diameter} or ${\cal P}$-{\em…

Let $G=\Sp(2g,\mathbb{Z})$ be the symplectic group over the integers. Given $m\in \mathbb{N}$, it is natural to ask if there exists a non-trivial matrix $A\in G$ such that $A^{m}=I$, where $I$ is the identity matrix in $G$. In this paper,…

A graph $G$ of order $n$ is called edge-pancyclic if, for every integer $k$ with $3 \leq k \leq n$, every edge of $G$ lies in a cycle of length $k$. Determining the minimum size $f(n)$ of a simple edge-pancyclic graph with $n$ vertices…

We prove an upper bound for geodesic periods of Maass forms over hyperbolic manifolds. By definition, such periods are integrals of Maass forms restricted to a special geodesic cycle of the ambient manifold, against a Maass form on the…

We determine a reasonable upper bound for the complexity of collection from the left to multiply two elements of a finite soluble, or polycyclic, group by restricting attention to certain polycyclic presentations of the group.

We provide an upper bound for the dimension of the maximal projective submodule of the Lie module of the symmetric group of $n$ letters in prime characteristic $p$, where $n = pk$ with $p \nmid k$.

We give an upper bound for the essential dimension of a smooth unipotent algebraic group over an arbitrary field. We also show that over a field $k$ which is finitely generated over a perfect field, a smooth unipotent algebraic $k$-group is…