Interpretation of the Arithmetic in certain groups of piecewise affine permutations of an interval

Permutations that Destroy Arithmetic Progressions in Elementary $p$-Groups

Given an abelian group $G$, it is natural to ask whether there exists a permutation $π$ of $G$ that "destroys" all nontrivial 3-term arithmetic progressions (APs), in the sense that $π(b) - π(a) \neq π(c) - π(b)$ for every ordered triple $(a,b,c) \in G^3$ satisfying $b-a = c-b \neq 0$. This question was resolved for infinite groups $G$ by Hegarty, who showed that there exists an AP-destroying permutation of $G$ if and only if $G/Ω_2(G)$ has the same cardinality as $G$, where $Ω_2(G)$ denotes the subgroup of all elements in $G$ whose order divides $2$. In the case when $G$ is finite, however, o…

A formal identity involving commuting triples of permutations

We prove a formal power series identity, relating the arithmetic sum-of-divisors function to commuting triples of permutations. This establishes a conjecture of Franklin T. Adams-Watters.

Realizations of groups of piecewise continuous transformations of the circle

We study the near action of the group PC of piecewise continuous self-transformations of the circle. Elements of this group are only defined modulo indeterminacy on a finite subset, which raises the question of realizability: a subgroup of PC is said to be realizable if it can be lifted to a group of permutations of the circle. We show that every finitely generated abelian subgroup of PC is realizable. We show that this is not true for arbitrary subgroups, by exhibiting a non-realizable finitely generated subgroup of the group of interval exchanges with flips. The group of (oriented) interval …

Permutation Groups with a Cyclic Two-Orbits Subgroup and Monodromy Groups of Siegel Functions

We classify the finite primitive permutation groups which have a cyclic subgroup with two orbits. This extends classical topics in permutation group theory, and has arithmetic consequences. By a theorem of C. L. Siegel, affine algebraic curves with infinitely many integral points are parametrized by rational functions whose monodromy groups have this property. We classify the possibilities of these monodromy groups, and give an application to Hilbert's irreducibility theorem.

Truncated versions of Dwork's lemma for exponentials of power series and $p$-divisibility of arithmetic functiens

(Dieudonné and) Dwork's lemma gives a necessary and sufficient condition for an exponential of a formal power series $S(z)$ with coefficients in $Q_p$ to have coefficients in $Z_p$. We establish theorems on the $p$-adic valuation of the coefficients of the exponential of $S(z)$, assuming weaker conditions on the coefficients of $S(z)$ than in Dwork's lemma. As applications, we provide several results concerning lower bounds on the $p$-adic valuation of the number of permutation representations of finitely generated groups. In particular, we give fairly tight lower bounds in the case of an arbi…

泛幻方排列与N元群

Panmagic permutations and N-ary groups

研究泛幻方排列的代数结构，发现其仿射子集在多元乘法下具有特殊性质，并与数论中的乘法阶、4k+1素数等经典主题相关联。

The rational function analogue of a question of Schur and exceptionality of permutation representations

In 1923 Schur considered the following problem. Let f(X) be a polynomial with integer coefficients that induces a bijection on the residue fields Z/pZ for infinitely many primes p. His conjecture, that such polynomials are compositions of linear and Dickson polynomials, was proved by M. Fried in 1970. Here we investigate the analogous question for rational functions, also we allow the base field to be any number field. As a result, there are many more rational functions for which the analogous property holds. Some infinite series come from rational isogenies of elliptic curves and deformations…

基于Post陪集定理的泛幻排列的N元群

N-ary groups of panmagic permutations from the Post coset theorem

研究泛幻排列（模n皇后问题的解）的代数结构，通过Post陪集定理证明其构成二面体子群陪集的N元群推广，并揭示与数论和组合学的联系。

FPn properties of generalized Houghton groups

This Thesis presents a 2-dimensional generalization of Houghtons' groups H_n. H_n is defined to be the group of all permutations p of a disjoint union of copies of the natural numbers N, with the property that each copy of N contains a cofinite subset on which p restricts to a translation. Our group G(n) is defined to be the group of all permutations p of a disjoint union of quadrants (i.e., copies of NxN) with the property that each quadrant contains a subquadrant on which p is a translation, while p restricted to the remaining set is piecewise isometric on a cofinite disjoint union of rays (…