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Let $G$ be a finite abelian group of exponent $n$, written additively, and let $A$ be a subset of $mathbb{Z}$. The constant $s_A(G)$ is defined as the smallest integer $ell$ such that any sequence over $G$ of length at least $ell$ has an $A$-weighted zero-sum of length $n$ and $eta_A(G)$ defined as the smallest integer $ell$ such that any sequence over $G$ of length at least $ell$ has an $A$-weighted zero-sum of length at most $n$. Here we prove that, for $alpha geq beta$, and $A=left{xinmathbb{N}; : ; 1 le a le p^{alpha} ; mbox{ and }; gcd(a, p) = 1right }$, we have $s_{A}(mathbb{Z}_{p^{alpha}}oplus mathbb{Z}_{p^beta}) = eta_A(mathbb{Z}_{p^{alpha}}oplus mathbb{Z}_{p^beta}) + p^{alpha}-1 = p^{alpha} + alpha +beta$ and classify all the extremal $A$-weighted zero-sum free sequences.
Let $G$ be a finite (not necessarily abelian) group and let $p=p(G)$ be the smallest prime number dividing $|G|$. We prove that $d(G)leq frac{|G|}{p}+9p^2-10p$, where $d(G)$ denotes the small Davenport constant of $G$ which is defined as the maximal
We prove a general stability theorem for $p$-class groups of number fields along relative cyclic extensions of degree $p^2$, which is a generalization of a finite-extension version of Fukudas theorem by Li, Ouyang, Xu and Zhang. As an application, we
In this paper, we study the surjectivity of adelic Galois representation associated to Drinfeld $mathbb{F}_q[T]$-modules over $mathbb{F}_q(T)$ of rank $2$ in the cases when $q$ is even or $q=3$.
We initiate an investigation of lattices in a new class of locally compact groups, so called locally pro-$p$-complete Kac-Moody groups. We discover that in rank 2 their cocompact lattices are particularly well-behaved: under mild assumptions, a cocom
As an analogue of a link group, we consider the Galois group of the maximal pro-$p$-extension of a number field with restricted ramification which is cyclotomically ramified at $p$, i.e, tamely ramified over the intermediate cyclotomic $mathbb Z_p$-e