Related papers: Units of $\mathbb{Z}C_{p^n}$

Let p be an odd prime. Let K = \Q(zeta) be the p-cyclotomic number field. Let v be a primitive root mod p and sigma : zeta --> zeta^v be a \Q-isomorphism of the extension K/\Q generating the Galois group G of K/\Q. For n in Z, the notation…

We consider the values taken by $n\times n$ circulant determinants with integer entries when $n$ is the product of two distinct odd primes $p,q$. These correspond to the integer group determinants for $\mathbb Z_{pq}$, the cyclic group of…

Suppose $C(G)$ denotes the set of all cyclic subgroups of a finite group $G$, and $\mathcal{O}_{2}(G)$ denotes the number of elements of order $2$ in $G$. In [Marius T., Finite groups with a certain number of cyclic subgroups. The American…

For a cyclic group $a$, define the atom of $a$ as the set of all elements generating $a$. Given any two elements $a,b$ of a finite cyclic group $G$, we study the sumset of the atom of $a$ and the atom of $b$. It is known that such a sumset…

We prove that there is an absolute constant $c>0$ with the following property: if $Z/pZ$ denotes the group of prime order $p$, and a subset $A\subset Z/pZ$ satisfies $1<|A|<p/2$, then for any positive integer…

We define and we characterize regular and c-regular cyclically ordered abelian groups. We prove that every dense c-regular cyclically ordered abelian group is elementarily equivalent to some cyclically ordered group of unimodular complex…

Let $P(x) \in \mathbb{Z}[x]$ be a polynomial. We give an easy and new proof of the fact that the set of primes $p$ such that $p \mid P(n)$, for some $n \in \mathbb{Z}$, is infinite. We also get analog of this result for some special…

We classify irreducible unitary representations of the group of all infinite matrices over a $p$-adic field ($p\ne 2$) with integer elements equipped with a natural topology. Any irreducible representation passes through a group $GL$ of…

For any positive integer $n$, let ${\rm C}_{n}$ be the cyclic group of order $n$. We determine all possible values of the integer group determinant of ${\rm C}_{2}^{2} \rtimes C_{4}$.

We give a list of finite groups containing all finite groups $G$ such that the group of units $\Z G^*$ of the integral group ring $\Z G$ is subgroup separable. There are only two types of these groups $G$ for which we cannot decide wether…

Let $p$ be a prime and $\mathbb{F}_p$ be a finite field of $p$ elements. Let $\mathbb{F}_pG$ denote the group algebra of the finite $p$-group $G$ over the field $\mathbb{F}_p$ and $V(\mathbb{F}_pG)$ denote the group of normalized units in…

Let $S := \{p_1,\ldots ,p_{\ell}\}$ be a finite set of primes and denote by $\mathcal{U}_S$ the set of all rational integers whose prime factors are all in $S$. Let $(U_n)_{n\geq 0}$ be a non-degenerate linear recurrence sequence with order…

An explicit formula for the Schur multiplier of the group of unitriangular matrices over products of cyclic rings $\ZZ/m\ZZ$ and $\ZZ$ is derived. We use it to provide presentations of the corresponding covering groups and touch upon the…

The Kronecker polynomial $K(z)$ is a finite product of cyclotomic polynomials $C_j(z)$. Any Kronecker polynomial $K(z)$ of degree $N+1$ with simple roots on the unit circle generates a finite set $\Phi_0=1, \Phi_1(z), \dots, \Phi_N(z) $ of…

In this article, we present an explicit formula for the $c$th nilpotent multiplier (the Baer invariant with respect to the variety of nilpotent groups of class at most $c\geq 1$) of the $n$th nilpotent product of some cyclic groups…

Let $\mathbb Z_n$ be the cyclic group of order $n \ge 3$ additively written. S. Savchev \& F. Chen (2007) proved that for each zero-sum free sequence $S = a_1 \bullet \dots \bullet a_t$ over $\mathbb Z_n$ of length $t > n/2$, there is an…

This paper provides two characterizations of the primitive roots of unity in quadratic cyclotomic extensions over arbitrary fields. Firstly, we introduce a mapping from $\mathbb{N}$ to $\mathbb{N}$ crucial for describing these roots,…

Let $G$ denote the projective special linear group $\text{PSL}(2,q)$, for a prime power $q$. It is shown that a finite 2-subgroup of the group $V(\mathbb{Z}G)$ of augmentation 1 units in the integral group ring $\mathbb{Z}G$ of $G$ is…

Given $n\geq 2$, $z_{ij}\in\mathbb{T}$ such that $z_{ij}=\overline z_{ji}$ for $1\leq i,j\leq n$ and $z_{ii}=1$ for $1\leq i\leq n$, and integers $p_1,...,p_n\geq 1$, we show that the universal $\mathrm{C}^*$-algebra generated by unitaries…

Our main result is elementary and concerns the relationship between the multiplicative groups of the coordinate and endomorphism rings of the formal additive group over a field of characteristic $p>0$. The proof involves the combinatorics…