A classification of two-generator 3-groups of second maximal class and low order is presented. All such groups with orders up to 3 are described, and. in some cases with orders up to 3The classification is based on computer aided computations. A description of the computations and their results are presented, together with an indication of their significance.Received 6 May 1977• We thank W.A. Alford and J.B. Ascione for writing parts of the computer program and especially thank M.F. Newman for many very helpful discussions.

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Minimal perfect hash functions are used for memory efficient storage and fast retrieval of items from static sets. We present an infinite family of efficient and practical algorithms for generating order preserving minimal perfect hash functions. We show that almost all members of the family construct space and time optimal order preserving minimal perfect hash functions, and we identify the one with minimum constants. Members of the family generate a hash function in two steps. First a special kind of function into an r−graph is computed probabilistically. Then this function is refined deterministically to a minimal perfect hash function. We give strong theoretical evidence that the first step uses linear random time. The second step runs in linear deterministic time. The family not only has theoretical importance, but also offers the fastest known method for generating perfect hash functions.

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Abstract. A recent form of the Todd-Coxeter algorithm, known as the lookahead algorithm, is described. The time and space requirements for this algorithm are shown experimentally to be usually either equivalent or superior to the Felsch and HaselgroveLeech-Trotter algorithms. Some findings from an experimental study of the behaviour of Todd-Coxeter programs in a variety of situations are given.1. Introduction. The Todd-Coxeter algorithm [20] (TC algorithm) is a systematic procedure for enumerating the cosets of a subgroup H of finite index in a group G, given a set of defining relations for G and words generating H. At the present time, Todd-Coxeter programs represent the most common application of computers to group theory. They are used for constructing sets of defining relations for particular groups, for determining the order of a group from its defining relations, for studying the structure of particular groups and for many other things.As an example of the use of the algorithm, consider the following family of defining relations, Men(n), due to Mennicke:

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Questions about nilpotency of groups satisfying Engel conditions have been considered since 1936, when Zorn proved that finite Engel groups are nilpotent. We prove that 4-Engel groups are locally nilpotent. Our proof makes substantial use of both hand and machine calculations.

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