Related papers: Multiplicative Subgroups of $\mathbb{Z}_p^*$ that …

We prove that for any prime $p$ and height $n \ge 1$, the telescopic Picard group $\mathrm{Pic}(\mathrm{Sp}_{Tn})$ contains a subgroup of the form $\mathbb{Z}_p \times \mathbb{Z}/a_p(p^n-1)$, where $a_p = 1$ if $p = 2$ and $a_p = 2$ if $p$…

We consider the integer group determinants for groups that are semidirect products of $\mathbb Z_p$ and $\mathbb Z_n$ with $p$ prime and $n\mid p-1$. We give a complete description of the integer group determinants for the general affine…

We prove a general combination theorem for discrete subgroups of $\mathrm{PGL}(n,\mathbb{R})$ preserving properly convex open subsets in the projective space $\mathbb{P}(\mathbb{R}^n)$, in the spirit of Klein and Maskit. We use it in…

By defining the classes of generalized co-Hopfian and relatively co-Hopfian groups, respectively, we consider two expanded versions of the generalized co-Bassian groups and of the classical co-Hopfian groups giving a close relationship with…

We prove some distribution results for the $k$-fold divisor function in arithmetic progressions to moduli that exceed the square-root of length $X$ of the sum, with appropriate constrains and averaging on the moduli, saving a power of $X$…

Let G be a group and let A be the algebra of complex functions on G with finite support. The product in G gives rise to a coproduct on A making it a multiplier Hopf algebra. In fact, because there exist integrals, we get an algebraic…

This paper presents formulae for the sum of the terms of a harmonic progression of order $k$ with integer parameters, $\mathrm{HP}_k(n)$, and for the partial sums of its two associated Fourier series, $C^z_{k}(a,b,n)$ and $S^z_{k}(a,b,n)$.…

Let $p$ be a prime. In this paper, we present a detailed $p$-adic analysis to factorials and double factorials and their congruences. We give good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number…

Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.

Let $p$ be a prime number and $\mathbb{Z}_p=\mathbb{Z}/p\mathbb{Z}$. We study finite groups with abelian derived subgroup and exponent $p$ in terms of group extension data and their matrix presentations. We show a one-to-one correspondence…

Let $p$ be an odd prime and $F_{\infty}$ a $p$-adic Lie extension of a number field $F$ with Galois group $G$. Suppose that $G$ is a compact pro-$p$ $p$-adic Lie group with no torsion and that it contains a closed normal subgroup $H$ such…

We show that every subset of SL_2(Z/pZ) grows rapidly when it acts on itself by the group operation. It follows readily that, for every set of generators A of SL_2(Z/pZ), every element of SL_2(Z/pZ) can be expressed as a product of at most…

Real points of Schottky space ${\mathcal S}_{g}$ are in correspondence with extended Kleinian groups $K$ containing, as a normal subgroup, a Schottky group $\Gamma$ of rank $g$ such that $K/\Gamma \cong {\mathbb Z}_{2n}$ for a suitable…

Let $p>5$ be a prime. We prove congruences modulo $p^{3-d}$ for sums of the general form $\sum_{k=0}^{(p-3)/2}\binom{2k}{k}t^k/(2k+1)^{d+1}$ and $\sum_{k=1}^{(p-1)/2}\binom{2k}{k}t^k/k^d$ with $d=0,1$. We also consider the special case…

We study whether a unital associative algebra $ A $ over a field admits a decomposition of the form $A = Z(A) + [A,A]$ where $ Z(A) $ is the center of $ A $ and $ [A,A] $ denotes the additive subgroup of $A$ generated by all additive…

We give an explicit list of all p-groups G of order at most p^4 or 2^5 such that the group algebra KG over the field K of characteristic p has a filtered multiplicative K-basis.

We give a self-contained proof of the fact that, for any prime number $p$, there exists a power series $$\Psi= \Psi_p(T) \in T + T^2\Z[[T]] $$ which trivializes the addition law of the formal group of Witt covectors is $p$-adically entire…

Let $\alpha$ and $\beta$ be two nonnegative integers such that $\beta < \alpha$. For an arbitrary sequence $\{a_n\}_{n\geqslant 1}$ of complex numbers, we consider the generalized Lambert series in order to investigate linear combinations…

In this paper, we introduce the power-partible reduction for holonomic (or, P-recursive) sequences and apply it to obtain a series of congruences for Ap\'ery numbers $A_k$. In particular, we prove that, for any $r\in\mathbb{N}$, there…

We fully classify completely multiplicative sequences which are given by generalised polynomial formulae, and obtain a similar result for (not necessarily completely) multiplicative sequences under the additional restriction that the…