  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.
