Given a finite group G and a prime p, a Sylow p-subgroup of G is a subgroup whose order is the largest power of p dividing |G|. Sylow's theorem asserts that such subgroups exist and that any two such are conjugate. This theorem is a fundamental tool in group-theoretic investigations. Similarly, in computational group theory there is an important role for efficient constructive analogues of Sylow's theorem. For computational purposes, it is typical to deal with large groups by specifying a permutation action, storing only a small set of generating permutations. This leads naturally to the following computational problems.

• SYLFIND(p, G)
  • Given: a prime p and generators for a group G,
  • Find: generators for a Sylow p-subgroup P ≤ G.
• SYLCONJ(P₁, P₂, G)
  • Given: generators for G and for two of its Sylow p-subgroups P₁ and P₂,
  • Find: an element g ∈ G for which P^g₁ = P₂.

In a recent series of papers, Kantor has shown that these problems have polynomial-time solutions. His methods exploit both a well-developed library of polynomial-time procedures for permutation groups and consequences of the classification of finite simple groups. The impressive nature of the machinery used for sequential solutions left open the question of whether there are methods that can take advantage of parallel machines. Following the prevalent paradigm, the question arises whether such problems are in the complexity class NC; i.e., are they solvable in polylogarithmic time (0(log^c n) steps) using a polynomial number of processors working in parallel? This dissertation establishes an affirmative answer, placing SYLFIND and SYLCONJ into NC.