Related papers: Terraces for Small Groups

There has been interest recently concerning when a left ordered group is locally indicable. Bergman and Tararin have shown that not all left ordered groups are locally indicable, but all known examples contain a nonabelian free subgroup. We…

A lattice-ordered group (an $\ell$-group) $G(\oplus, \vee, \wedge)$ can be naturally viewed as a semiring $G(\vee,\oplus)$. We give a full classification of (abelian) $\ell$-groups which are finitely generated as semirings, by first showing…

We enumerate the 15768 perfect groups of order up to $2\cdot 10^6$, up to isomorphism, thus also completing the missing cases in the prior classification. The work supplements the by now well-understood computer classifications of solvable…

We propound the thesis that there is a limitation to the number of possible structures which are axiomatically endowed with identities involving operations. In the case of algebras with a binary operation satisfying a formally reducible (to…

We study the structure of the algebraic fundamental group for minimal surfaces of general type S satisfying K_S^2<=3\chi-2$ and not having any irregular etale cover. We show that, if K_S^2<=3\chi-2, then then the algebraic fundamental group…

We show that for any natural number $s$, there is a constant $\gamma$ and a subgraph-closed class having, for any natural $n$, at most $\gamma^n$ graphs on $n$ vertices up to isomorphism, but no adjacency labeling scheme with labels of size…

Let G be a finite abelian group with |G|>1. Let a_1,...,a_k be k distinct elements of G and let b_1,...,b_k be (not necessarily distinct) elements of G, where k is a positive integer smaller than the least prime divisor of |G|. We show that…

A positive integer $n$ is defined to be cyclic if and only if every group of size $n$ is cyclic. Equivalently, $n$ is cyclic if and only if $n$ is relatively prime to the number of positive integers less than $n$ that are relatively prime…

We present a construction of a Jordan scheme from an elementary abelian $2$-group of rank $n$ and a $\{1,-1\}$-matrix of order $2^n$ that satisfies a specified condition. We then prove that the orders of matrices with the specified…

A latin square of order $n$ with pairwise disjoint subsquares of orders $h_1,\dots,h_k$ such that $h_1+\dots+h_k = n$ is known as a realization. The existence of realizations is a partially solved problem with a few general results for an…

For an abelian variety $A$ over a number field $F$, we prove that the average rank of the quadratic twists of $A$ is bounded, under the assumption that the multiplication-by-3 isogeny on $A$ factors as a composition of 3-isogenies over $F$.…

The Main Theorem for abelian fields (often called Main Conjecture despite proofs in most cases) has a long history which has found a solution by means of "elementary arithmetic", as detailed in Washington's book from Thaine's method having…

Given a sequence (a_k) = a_0, a_1, a_2,... of real numbers, define a new sequence L(a_k) = (b_k) where b_k = a_k^2 - a_{k-1} a_{k+1}. So (a_k) is log-concave if and only if (b_k) is a nonnegative sequence. Call (a_k) "infinitely…

A well-known conjecture asserts that the mapping class group of a surface (possibly with punctures/boundary) does not virtually surject onto $\Z$ if the genus of the surface is large. We prove that if this conjecture holds for some genus,…

A $k$-plex in a latin square of order $n$ is a selection of $kn$ entries that includes $k$ representatives from each row and column and $k$ occurrences of each symbol. A $1$-plex is also known as a transversal. It is well known that if $n$…

A relation on a $k$-net$(n)$ (or, equivalently, a set of $k-2$ mutually orthogonal Latin squares of order $n$) is an $\mathbb{F}_{2}$ linear dependence within the incidence matrix of the net. Dukes and Howard (2014) showed that any…

The notion of formal duality in finite Abelian groups appeared recently in relation to spherical designs, tight sphere packings, and energy minimizing configurations in Euclidean spaces. For finite cyclic groups it is conjectured that there…

Lower bounds for the number of local nearrings on groups of order $p^3$ are obtained. On each non-metacyclic non-abelian or metacyclic abelian groups of order $p^3$ there exist at least $p+1$ non-isomorphic local nearrings

In the paper "An Abelian Loop for Non-Composites" (arXiv:110.14716), we introduced a group-like structure consisting of odd prime numbers and 1, with properties that allowed us to prove analogous results to well known theorems in Number…

A Latin square has six conjugate Latin squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a Latin square has conjugate symmetry if at least two of its six conjugates are equal. We enumerate Latin squares…