Let $G$ be a finite group given as input by its multiplication table. For a subset $S$ of $G$ and an element $g\in G$ the Cayley Group Membership Problem (denoted CGM) is to check if $g$ belongs to the subgroup generated by $S$. While this problem is easily seen to be in polynomial time, pinpointing its parallel complexity has been of research interest over the years. In this paper we further explore the parallel complexity of the abelian CGM problem, with focus on the dynamic setting: the generating set $S$ changes with insertions and deletions and the goal is to maintain a data structure that supports efficient membership queries to the subgroup $\angle{S}$. We obtain the following results: 1. We first consider the more general problem of Monoid Membership. When $G$ is a commutative monoid we give a deterministic dynamic algorithm constant time parallel algorithm for membership testing that supports $O(1)$ insertions and deletions in each step. 2. Building on the previous result we show that there is a dynamic randomized constant-time parallel algorithm for abelian CGM that supports polylogarithmically many insertions/deletions to $S$ in each step. 3. If the number of insertions/deletions is at most $O(\log n/\log\log n)$ then we obtain a deterministic dynamic constant-time parallel algorithm for the problem. 4. We obtain analogous results for the dynamic abelian Group Isomorphism.

翻译：设$G$是一个有限群，其乘法表作为输入给出。对于子集$S\subseteq G$及元素$g\in G$，凯莱群成员问题（记为CGM）是检查$g$是否属于由$S$生成的子群。尽管此问题显然属于多项式时间可解问题，但其并行复杂度的精确定位多年来一直是研究兴趣所在。本文进一步探讨阿贝尔CGM问题的并行复杂度，重点关注动态设置：生成集$S$通过插入和删除操作发生变化，目标是维护一个支持对子群$\angle{S}$进行高效成员查询的数据结构。我们获得了以下结果：1. 首先考虑更一般的幺半群成员问题。当$G$为交换幺半群时，我们给出一个确定性动态常数时间并行算法，用于成员测试，该算法每一步支持$O(1)$次插入和删除操作。2. 基于前述结果，我们证明存在一个动态随机化常数时间并行算法，用于阿贝尔CGM问题，该算法每一步支持对$S$进行多对数次插入/删除操作。3. 若插入/删除次数不超过$O(\log n/\log\log n)$，则可获得该问题的确定性动态常数时间并行算法。4. 对于动态阿贝尔群同构问题，我们得到了类似结果。