Sylow's theorem is a fundamental tool in group-theoretic investigations. In computational group theory there is an important role for efficient constructive analogs of Sylow's theorem. For computational purposes, we assume that a group is given by a set of permutations that generate it. This leads to the following problems:

SYLFIND(p,G)

• GIVEN: a prime p and generators for a group G,
• FIND: generators for a Sylow p-subgroup P ≤ G.

SYLCONJ(P1₁ P₂, G)

• GIVEN: generators for G and two of its Sylow p-subgroups P₁ and P₂,
• FIND: an element g ∈ G for which P₁^g = P₂.

SYLEMBED(P, G)

• GIVEN: generators for G and for a p-subgroup P ≤ G,
• FIND: generators for a Sylow p-subgroup of G containing P.

This paper shows SYLFIND, SYLCONJ, and SYLEMBED for solvable groups are in the complexity class NC; namely, they are solvable in poly-logarithmic time (O(log^c n) steps) using a polynomial number of processors working in parallel. A future paper by Kantor, Luka, and Mark extends these results to general groups.