Related papers: Generating Sets of Mathieu Groups
We construct a 2-generator recursively presented group with infinite torsion length. We also explore the construction in the context of solvable and word-hyperbolic groups.
We discuss the anomalous U(1) gauge symmetry as a mechanism of generating the grand-unification scale. We conclude that unification to a simple group cannot be realized unless some parameters are ``tuned'', and that models with product…
A natural generating set for a Galois extension regarded as the splitting field of an irreducible polynomial is introduced and investigated here. Minimal generating sets arising in this context throw many surprises compared to the analogous…
We study graded rings of modular forms over congruence subgroups, with coefficients in a subring $A$ of $\mathbb{C}$, and specifically the highest weight needed to generate these rings as $A$-algebras. In particular, we determine upper…
In this paper, we introduce several notions of "dimension" of a finite group, involving sizes of generating sets and certain configurations of maximal subgroups. We focus on the inequality $m(G) \leq \mathrm{MaxDim}(G)$, giving a family of…
For positive integers $n$ and $m$, consider a multiset of non-empty subsets of $[m]$ such that there is a \textit{unique} partition of these subsets into $n$ partitions of $[m]$. We study the maximum possible size $g(n,m)$ of such a…
Mullineux defined an involution on the set of $e$-regular partitions of $n$. When $e=p$ is prime, these partitions label irreducible symmetric group modules in characteristic $p$. Mullineux's conjecture, since proven, was that this…
We study how the spectral gap and diameter of Cayley graphs depend strongly on the choice of generating set. We answer a question of Pyber and Szab\'o (2013) by exhibiting a sequence of finite groups $G_n$ with $|G_n| \to \infty$ admitting…
Given an algebra $F[H]^G$ of polynomial invariants of an action of the group $G$ over the vector space $H$, a subset $S$ of $F[H]^G$ is called separating if $S$ separates all orbits that can be separated by $F[H]^G$. A minimal separating…
We introduce the depth parameters of a finite semigroup, which measure how hard it is to produce an element in the minimum ideal when we consider generating sets satisfying some minimality conditions. We estimate such parameters for some…
We consider the (pure) braid groups B_{n}(M) and P_{n}(M), where M is the 2-sphere S^2 or the real projective plane RP^2. We determine the minimal cardinality of (normal) generating sets X of these groups, first when there is no restriction…
The problem of bounding the size of a set system under various intersection restrictions has a central place in extremal combinatorics. We investigate the maximum number of disjoint pairs a set system can have in this setting. In…
In order to better understand the structure of closed collections of reversible gates, we investigate the lattice of closed sets and the maximal members of this lattice. In this note, we find the maximal closed sets over a finite alphabet.…
A subset of the Hamming cube over $n$-letter alphabet is said to be $d$-maximal if its diameter is $d$, and adding any point increases the diameter. Our main result shows that each $d$-maximal set is either of size at most $(n+o(n))^d$ or…
The reasons behind the gauge symmetry of the Standard Model, U(1)xSU(2)xSU(3), are still unsettled. One obvious feature is the low dimensionality of all its subgroups. Under certain conditions, a negative answer to the question "why not…
We study the number $M_n(T)$ be the number of integer $n\times n$ matrices $A$ with entries bounded in absolute value by $T$ such that the Galois group of characteristic polynomial of $A$ is not the full symmetric group $S_n$. One knows…
In 1934, Garrett Birkhoff has shown that the number of isomorphism classes of finite metabelian groups of order $p^{22}$ tends to infinity with $p$. More precisely, for each prime number $p$ there is a family…
We consider a very simple Mealy machine (three states over a two-symbol alphabet), and derive some properties of the semigroup it generates. In particular, this is an infinite, finitely generated semigroup; we show that the growth function…
A finite subset $X$ of the Euclidean space is called an $m$-distance set if the number of distances between two distinct points in $X$ is equal to $m$. An $m$-distance set $X$ is said to be maximal if any vector cannot be added to $X$ while…
A generating set $S$ for a group $G$ is independent if the subgroup generated by $S\setminus \{s\}$ is properly contained in $G$, for all $s \in S.$ In this paper, we study a problem proposed by Peter Glasby: we investigate finite groups,…