FAST MANAGEMENT OF PERMUTATION-GROUPS .1.

We present new algorithms for permutation group manipulation. Our meth ods result in an improvement of nearly an order of magnitude in the wo rst-case analysis for the fundamental problems of finding strong gener ating sets and testing membership. The normal structure of the group i s brought into play even for such elementary issues. An essential elem ent is the recognition of large alternating composition factors of the given group and subsequent extension of the permutation domain to dis play the natural action of these alternating groups. Further new featu res include a novel fast handling of alternating groups and the siftin g of defining relations in order to link these and other analyzed fact ors with the rest of the group. The analysis of the algorithm depends on the classification of finite simple groups. In a sequel to this pap er, using an enhancement of the present method, we shall achieve a fur ther order of magnitude improvement.