The authors have used a wide variety of examples to demonstrate the effectiveness of new algorithms for the computation of Sylow subgroups. The effectiveness of these algorithms makes it possible to compute Sylow functions for considerably larger orders and degrees than heretofore feasible. The paper is highly recommended for those engaged in computation of Sylow functions or advancement of available techniques in that area.

The paper is clearly written and well organized in four sections. Section 1 provides a critical discussion of the available algorithms for computing Sylow functions. Section 2 presents the derivation of the algorithm for finding a Sylow p-subgroup of a permutation group for a given prime p. The O’Nan-Scott theorem is the basis of the derivations. Altogether, five input conditions have been considered. In section 3, the algorithm for finding a conjugating element for two given Sylow p-subgroups is described for these conditions. Finally, in section 4, the authors present the comparative computer CPU times for computing Sylow functions for more than 120 cases of varying orders and degrees.