We discuss aspects of computation in permutation groups assuming polynomial time ae a measure of efficiency. Of particular interest are problems, such as finding the Intersection of two groups, that resemble or generalize the problem of testing graph Isomorphism. We also summarize the instances where the problems are known to be solvable In polynomial time and indicate methods that accomplish this. As with graph Isomorphism, the computational complexities of the general problems are open, though we can demonstrate polynomial-time reductions and equivalences among them. A typical approach to such issues is shown to involve an NP-complete problem. Several open questions are listed.