Dedicated to Benjamin Fine on the occasion of his 60th birthday. The object of this paper is to give a proof of the following theorem: S/P ∼ = Λ mn ⊆ Q + , where S/P is a certain torsion-free factor group of the Baumslag-Solitar group a, b; a −1 b m a = b n | m = 0, n = 0, m, n ∈ Z, with m and n are relatively prime, and Λ mn is a subgroup of the additive group of the rational numbers Q + .

let r be a commutative ring with identity and m be a unitary r-module. let : s(m) −! s(m) [ {;} be a function, where s(m) is the set of submodules ofm. suppose n  2 is a positive integer. a proper submodule p of m is called(n − 1, n) − -prime, if whenever a1, . . . , an−1 2 r and x 2 m and a1 . . . an−1x 2p(p), then there exists i 2 {1, . . . , n − 1} such that a1 . . . ai−1ai+1 . . . an−1x...

Let $E$ be an elliptic curve, with identity $O$, and let $C$ a cyclic subgroup of odd order $N$, over algebraically closed field $k$ $\operatorname{char} k \nmid N$. For $P \in C$, $s_P$ rational function divisor $N \cdot P - N O$. We ask whether the $N$ functions are linearly independent. generic $(E,C)$, we prove that answer is yes. bound number exceptional $(E,C)$ when prime by using geometr...

On primitive roots of 1 mod p k , divisors of p ± 1, Wieferich primes, and quadratic analysis mod p Abstract Primitive roots of 1 mod p k (k > 2 and odd prime p) are sought, in cyclic units group G k ≡ A k B k mod p k , coprime to p, of order (p − 1)p k−1. 'Core' subgroup A k has order p − 1 independent of precision k, and 'extension' subgroup B k of all p k−1 residues 1 mod p is generated by p...

An Engel sink of an element g a group G is set $${\cal E}(g)$$ such that for every x ∈ all sufficiently long commutators [⋯[[x, g], g],…, g] belong to . (Thus, precisely when we can choose E}(g) = \{ 1\} $$ .) It proved if profinite admits elementary abelian automorphisms A coprime order q2 prime q each {1} the centralizer CG(a) has countable (or finite) sink, then finite normal subgroup N G/N ...

Let $A$ be a Noetherian ring, $I$ be an ideal of $A$ and $sigma$ be a semi-prime operation, different from the identity map on the set of all ideals of $A$. Results of Essan proved that the sets of associated prime ideals of $sigma(I^n)$, which denoted by $Ass(A/sigma(I^n))$, stabilize to $A_{sigma}(I)$. We give some properties of the sets $S^{sigma}_{n}(I)=Ass(A/sigma(I^n))setminus A_{sigma}(I...

In this paper we study a geometric coding algorithm for indefinite binary quadratic forms Q for the congruence subgroup Γ(N), with respect to the usual fundamental domain FN , where N is assumed prime. The cycles Q1, . . . , Qn that this algorithm produces are such that the the corresponding paths γ1, . . . , γn in the Riemann surface X0(N)(C) have a nice behaviour around the elliptic points of...

We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in Zp , whenever p is a fixed prime. For the induction step, we introduce the ...

Abstract In this paper, we call a finite group G G an N L M NLM -group ( C NCM -group, respectively) if every non-normal cyclic subgroup of prime order or 4 (prime power order, in is contained maximal . Using the property -groups and -groups, give new necessary suffic...

We show that every weighing matrix of weight n invariant under a finite abelian group G can be generated from a subgroup H of G with |H| ≤ 2n−1. Furthermore, if n is an odd prime power and a proper circulant weighing matrix of weight n and order v exists, then v ≤ 2n−1. We also obtain a lower bound on the weight of group invariant matrices depending on the invariant factors of the underlying