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Jun Seok Oh

Publications and source records attributed to Jun Seok Oh.

12 recordsLinked to original sources
arXiv2026

The catenary degree of monoids of product-one sequences

Let $G$ be a (multiplicatively written) finite group. A sequence over $G$ is a finite collection of terms from $G$, where repetition is allowed and the order is disregarded. A product-one sequence is a sequence whose terms can be ordered such that their product in $G$ equals the identity element of $G$. The set $\mathcal B (G)$ of all product-one sequences over $G$, endowed with the concatenation of sequences as the operation, is a finitely generated C-monoid; in particular, it is atomic, i.e., every non-unit element can be written as a finite product of atoms. The study of $\mathcal B (G)$ is of fundamental importance, as its combinatorial, algebraic, and arithmetic properties play a crucial role across various branches of mathematics, most notably in invariant theory and factorization theory. While the arithmetic of the monoid $\mathcal B (G)$ is well understood in the abelian setting (in which case $\mathcal B (G)$ is a Krull monoid), little is known in the non-abelian setting because of the substantial structural complexity involved. In this paper, we study the arithmetic invariants of the monoid $\mathcal B (G)$ for non-abelian groups, focusing in particular on the catenary degree. The catenary degree $\mathsf c (G)$ of the monoid $\mathcal B (G)$ is defined as the smallest integer $N$ such that any two factorizations of an element $S \in \mathcal B (G)$ can be concatenated by a chain of factorizations in which adjacent steps differ by replacing at most $N$ atoms. Extending the methods from arithmetic combinatorics to the non-abelian setting, we explicitly characterize all finite groups with catenary degree at most 3, and we investigate an infinite class of finite groups whose monoids of product-one sequences are seminormal and possess well-behaved arithmetic structures. Furthermore, we show that a specific non-abelian group in this class has catenary degree 4.

math.GR↗
arXiv2026

On algebraic and arithmetic properties of monoids of product-$K$ sequences

Let $G$ be a group and $K$ be a normal subgroup of $G$. A sequence over $G$ is a finite collection of terms from $G$, where repetition is allowed, and the order is disregarded. A product-$K$ sequence is a sequence whose terms can be ordered such that their product in $G$ belongs to $K$. The set $\mathcal B_K (G)$ of all product-$K$ sequences over $G$ forms a monoid, called the monoid of product-$K$ sequences, under the operation of sequence concatenation. In this paper, we investigate the algebraic and arithmetic properties of the monoid $\mathcal B_K (G)$. Among our main results, we provide precise characterizations of when the monoid $\mathcal B_K (G)$ satisfies key properties, namely being a (transfer) Krull, seminormal, or (half-)factorial. Our results generalize existing frameworks, making them applicable to both the classical abelian and the more recently developed non-abelian settings.

math.GR↗
arXiv2026

On zero-sum problems over metacyclic groups $C_n \rtimes_s C_2$

Let $G$ be a finite group. A finite collection of elements from $G$, where the order is disregarded and repetitions are allowed, is said to be a product-one sequence if its elements can be ordered such that their product in $G$ equals the identity element of $G$. Then, the Gao's constant $\mathsf E (G)$ of $G$ is the smallest integer $\ell$ such that every sequence of length at least $\ell$ has a product-one subsequence of length $|G|$. For a positive integer $n$, we denote by $C_n$ a cyclic group of order $n$. Let $G = C_n \rtimes_s C_2$ with $s^2\equiv 1\pmod n$ be a metacyclic group. The direct and inverse problems of $\mathsf E (G)$ were settled recently, except for the case that $G=C_{3n_2}\rtimes_s C_2$ with $n_2\neq 1$, $\gcd(n_2,6)=1$, $s\equiv -1 \pmod 3$, and $s\equiv 1\pmod {n_2}$. In this paper, we complete the remaining case and hence for all metacyclic groups of the form $G=C_n \rtimes C_2$, the Gao's constant and the associated inverse problem are now fully settled (see Theorem 1.2).

math.CO↗
arXiv2025

A classification of finite groups with small Davenport constant

Let $G$ be a finite group. By a sequence over $G$, we mean a finite unordered string of terms from $G$ with repetition allowed, and we say that it is a product-one sequence if its terms can be ordered so that their product is the identity element of $G$. Then, the Davenport constant $\mathsf D (G)$ is the maximal length of a minimal product-one sequence, that is a product-one sequence which cannot be factored into two non-trivial product-one subsequences. The Davenport constant is a combinatorial group invariant that has been studied fruitfully over several decades in additive combinatorics, invariant theory, and factorization theory, etc. Apart from a few cases of finite groups, the precise value of the Davenport constant is unknown. Even in the abelian case, little is known beyond groups of rank at most two. On the other hand, for a fixed positive integer $r$, structural results characterizing which groups $G$ satisfy $\mathsf D (G) = r$ are rare. We only know that there are finitely many such groups. In this paper, we study the classification of finite groups based on the Davenport constant.

math.GR↗
arXiv2023

Prime Factorization of ideals in commutative rings, with a focus on Krull rings

Let $R$ be a commutative ring with identity. The structure theorem says that $R$ is a PIR (resp., UFR, general ZPI-ring, $π$-ring) if and only if $R$ is a finite direct product of PIDs (resp., UFDs, Dedekind domains, $π$-domains) and special primary rings. All of these four types of integral domains are Krull domains, so motivated by the structure theorem, we study the prime factorization of ideals in a ring that is a finite direct product of Krull domains and special primary rings. Such a ring will be called a general Krull ring. It is known that Krull domains can be characterized by the star operations $v$ or $t$ as follows: An integral domain $R$ is a Krull domain if and only if every nonzero proper principal ideal of $R$ can be written as a finite $v$- or $t$-product of prime ideals. However, this is not true for general Krull rings. In this paper, we introduce a new star operation $u$ on $R$, so that $R$ is a general Krull ring if and only if every proper principal ideal of $R$ can be written as a finite $u$-product of prime ideals. We also study several ring-theoretic properties of general Krull rings including Kaplansky-type theorem, Mori-Nagata theorem, Nagata rings, and Noetherian property.

math.AC↗
arXiv2020

On product-one sequences over dihedral groups

Let $G$ be a finite group. A sequence over $G$ means a finite sequence of terms from $G$, where repetition is allowed and the order is disregarded. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. The set of all product-one sequences over $G$ (with concatenation of sequences as the operation) is a finitely generated C-monoid. Product-one sequences over dihedral groups have a variety of extremal properties. This article provides a detailed investigation, with methods from arithmetic combinatorics, of the arithmetic of the monoid of product-one sequences over dihedral groups.

math.NT↗
arXiv2020

On zero-sum free sequences contained in random subsets of finite cyclic groups

Let $C_n$ be a cyclic group of order $n$. A sequence $S$ of length $\ell$ over $C_n$ is a sequence $S = a_1\boldsymbol\cdot a_2\boldsymbol\cdot \ldots\boldsymbol\cdot a_{\ell}$ of $\ell$ elements in $C_n$, where a repetition of elements is allowed and their order is disregarded. We say that $S$ is a zero-sum sequence if $Σ_{i=1}^{\ell} a_i = 0$ and that $S$ is a zero-sum free sequence if $S$ contains no zero-sum subsequence. Let $R$ be a random subset of $C_n$ obtained by choosing each element in $C_n$ independently with probability $p$. Let $N^R_{n-1-k}$ be the number of zero-sum free sequences of length $n-1-k$ in $R$. Also, let $N^R_{n-1-k,d}$ be the number of zero-sum free sequences of length $n-1-k$ having $d$ distinct elements in $R$. We obtain the expectation of $N^R_{n-1-k}$ and $N^R_{n-1-k,d}$ for $0\leq k\leq \big\lfloor \frac{n}{3} \big\rfloor$. We also show a concentration result on $N^R_{n-1-k}$ and $N^R_{n-1-k,d}$ when $k$ is fixed.

math.CO↗
arXiv2019

On Erdős-Ginzburg-Ziv inverse theorems for Dihedral and Dicyclic groups

Let $G$ be a finite group and exp$(G)$ = lcm$\{$ord$(g)$$\mid$$g \in G \}$. A finite unordered sequence of terms from $G$, where repetition is allowed, is a product-one sequence if its terms can be ordered such that their product equals the identity element of $G$. We denote by $\mathsf s (G)$ (or $\mathsf E (G)$ respectively) the smallest integer $\ell$ such that every sequence of length at least $\ell$ has a product-one subsequence of length $\exp (G)$ (or $|G|$ respectively). In this paper, we provide the exact values of $\mathsf s (G)$ and $\mathsf E (G)$ for Dihedral and Dicyclic groups and we provide explicit characterizations of all sequences of length $\mathsf s (G) - 1$ (or $\mathsf E (G) - 1$ respectively) having no product-one subsequence of length $\exp (G)$ (or $|G|$ respectively).

math.CO↗
arXiv2019

On minimal product-one sequences of maximal length over Dihedral and Dicyclic groups

Let $G$ be a finite group. By a sequence over $G$, we mean a finite unordered sequence of terms from $G$, where repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered such that their product equals the identity element of $G$. The large Davenport constant $\mathsf D (G)$ is the maximal length of a minimal product-one sequence, that is, a product-one sequence which cannot be factored into two non-trivial product-one subsequences. We provide explicit characterizations of all minimal product-one sequences of length $\mathsf D (G)$ over Dihedral and Dicyclic groups. Based on these characterizations we study the unions of sets of lengths of the monoid of product-one sequences over these groups.

math.CO↗
arXiv2018

On the Algebraic and Arithmetic structure of the monoid of Product-one sequences II

Let $G$ be a finite group and $G'$ its commutator subgroup. By a sequence over $G$, we mean a finite unordered sequence of terms from $G$, where repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered such that their product equals the identity element of $G$. The monoid $\mathcal B (G)$ of all product-one sequences over $G$ is a finitely generated C-monoid whence it has a finite commutative class semigroup. It is well-known that the class semigroup is a group if and only if $G$ is abelian (equivalently, $\mathcal B (G)$ is Krull). In the present paper we show that the class semigroup is Clifford (i.e., a union of groups) if and only if $|G'| \le 2$ if and only if $\mathcal B (G)$ is seminormal, and we study sets of lengths in $\mathcal B (G)$.

math.AC↗
arXiv2018

On the algebraic and arithmetic structure of the monoid of product-one sequences

Let $G$ be a finite group. A finite unordered sequence $S = g_1 \boldsymbol{\cdot} \ldots \boldsymbol{\cdot} g_{\ell}$ of terms from $G$, where repetition is allowed, is a product-one sequence if its terms can be ordered such that their product equals $1_G$, the identity element of the group. As usual, we consider sequences as elements of the free abelian monoid $\mathcal F (G)$ with basis $G$, and we study the submonoid $\mathcal B (G) \subset \mathcal F (G)$ of all product-one sequences. This is a finitely generated C-monoid, which is a Krull monoid if and only if $G$ is abelian. In case of abelian groups, $\mathcal B (G)$ is a well-studied object. In the present paper we focus on non-abelian groups, and we study the class semigroup and the arithmetic of $\mathcal B (G)$.

math.AC↗