Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P , the subgroup Cl1(ZP ) of SK1(ZP ), in terms of a genetic basis of P . We also introduce a deflation map Cl1(ZP ) → Cl1 ( Z(P/N) ) , for a normal subgroup N of P , and show that it is always surjective. Along the way, we give a new ...

Suppose $p \geq 5$ is a prime number, and let $G = \textrm{SL}_2(\mathbb{Q}_p)$. We calculate the derived functors $\textrm{R}^n\mathcal{R}_B^G(\pi)$, where $B$ Borel subgroup of $G$, $\mathcal{R}_B^G$ right adjoint smooth parabolic induction constructed by Vign\'eras, $\pi$ any smooth, absolutely irreducible, mod $p$ representation $G$.

We give efficient quantum algorithms for HIDDEN TRANSLATION and HIDDEN SUBGROUP in a large class of non-abelian groups including solvable groups of bounded exponent and of bounded derived series. Our algorithms are recursive. For the base case, we solve efficiently HIDDEN TRANSLATION in Z p , whenever p is a fixed prime. For the induction step, we introduce the problem ORBIT COSET generalizing ...

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...