Constructive Sylow Theorems for the classical groups

Let r be a prime. We provide efficient algorithms to construct and solve the conjugacy problem for the Sylow r-subgroups of the classical groups over finite fields in their natural representations. We also provide algorithms to construct the normaliser of a Sylow r-subgroup of GL(n,Fq), Sp(2m,Fq), GO(2m+1,Fq), GO−(2m,Fq), GO+(2m,Fq) or GU(n,Fq2). The algorithms described here are implemented in the Magma Computer Algebra System.