Related papers: Harada's conjecture II and Gramian determinants

A critical value on an abelian group G of odd order d is a value $\lambda$ such that the functional equation f$\star$f (2 t) = $\lambda$f (t)^2 on G has a nonzero solution f. We construct many critical values by using abelian varieties with…

Let $\pi$ be a set of primes such that $|\pi|\geqslant 2$ and $\pi$ differs from the set of all primes. Denote by $r$ the smallest prime which does not belong to $\pi$ and set $m=r$ if $r=2,3$ and $m=r-1$ if $r\geqslant 5$. We study the…

We connect the existence of a ternary classical universal quadratic form over a totally real number field $K$ with the property that all totally positive multiples of 2 are sums of squares (if $K$ does not contain $\sqrt 2$ or contains a…

We construct an infinite word $w$ over the $5$-letter alphabet such that for every factor $f$ of $w$ of length at least two, there exists a cyclic permutation of $f$ that is not a factor of $w$. In other words, $w$ does not contain a…

Based on the Goldbach conjecture and arithmetic fundamental theorem, the Goldbach conjecture was extended to more general situations, i.e., any positive integer can be written as summation of some specific prime numbers, which depends on…

Artin's Conjecture on Primitive Roots states that a non-square nonunit integer $a$ is a primitive root modulo $p$ for the positive proportion of $p$. This conjecture remains open, but on average, there are many results due to P. J.…

We give a definition of the Yagita invariant at a prime p of an arbitrary group G, and compute the invariant for each prime for the general linear groups over any integrally closed subring of the complex numbers. We also compute the…

A non-trivial element of a group is a generalized torsion element if some products of its conjugates is the identity. The minimum number of such conjugates is called a generalized torsion order. We provide several restrictions for…

Recently Ritter and Weiss introduced an equivariant "main conjecture" than generalizes and refines the Main Conjecture of Iwasawa theory. In this paper, we show that, for the prime 2 and a dihedral extension of order 8 over Q, this…

An element $g$ of a group is called reversible if it is conjugate in the group to its inverse. An element is an involution if it is equal to its inverse. This paper is about factoring elements as products of reversibles in the group…

Let $H$ be an extension of a finite group $Q$ by a finite group $G$. Inspired by the results of duality theorems for \'etale gerbes on orbifolds, we describe the number of conjugacy classes of $H$ that maps to the same conjugacy class of…

Let $G$ be a finite group, and let $d$ be the degree of an irreducible character of $G$ such that $|G|=d(d+e)$ for some $e>1$. Consider the case when $G$ is solvable, $d$ is square-free, and $(d,d+e)=1$. We wish to explore an equivalent…

The aim of this paper is to give a full detail of the proof given by Harder of a theorem on the denominator of the Eisenstein class for $\mathrm{SL}_2(\mathbb{Z})$ and to show that the theorem has some interesting applications including the…

We give a simple combinatorial proof of the $\lambda_g$ conjectue in genus 2. We use a description of the class $\lambda_2$ as a linear combination of boundary strata, and show the conjecture follows inductively from applications of the…

In this paper, we focus on Narayana numbers which can be written as a products of four repdigits in base $g$, where $g$ is an integer with $g\geq2$. We prove that for $g$ between $2$ and $12$, there are finitely many of these numbers.…

A necessary condition for uniqueness of factorizations of elements of a finite group $G$ with factors belonging to a union of some conjugacy classes of $G$ is given. This condition is sufficient if the number of factors belonging to each…

The Kida's formula in classical Iwasawa theory relates the Iwasawa $\lambda$-invariants of $p$-extensions of number fields. Analogue of this formula was subsequently established for the Iwasawa $\lambda$-invariants of Selmer groups under an…

Let $\chi$ be an irreducible character of a finite group $G$. A. R. Miller conjectured that the proportion of elements $g\in G$ such that $\chi(g)$ is zero or a root of unity is at least 1/2. We construct a character of a perfect group of…

An $integral$ of a group $G$ is a group $H$ whose derived group (commutator subgroup) is isomorphic to $G$. This paper discusses integrals of groups, and in particular questions about which groups have integrals and how big or small those…

It is well known that the number of real irreducible characters of a finite group G coincides with the number of real conjugacy classes of G. Richard Brauer has asked if the number of irreducible characters with Frobenius-Schur indicator 1…