**TITLE:** Lattices with sublattices of a given order

**AUTHORS:** George Havas and Martin Ward

**CITATION:** Journal of Combinatorial Theory **7** (1969) 281-282 (open archive)

**ABSTRACT:** Birkhoff in his 1948 edition of *Lattice Theory* poses the following problem. Given n, what is the smallest integer f(n) such that every lattice with order r >= f(n) elements contains a sublattice of exactly n elements? We show that f(n) exists for each n > 0 and f(n) < n^{3n} for n > 1.

**TITLE:** Implementation and analysis of the Todd-Coxeter algorithm

**AUTHORS:** John J. Cannon, Lucien A. Dimino, George Havas and Jane M. Watson

**CITATION:** Mathematics of Computation 27 (1973) 463-490 (free archive)

**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 Haselgrove-Leech-Trotter algorithms. Some findings from an experimental study of the behaviour of Todd-Coxeter programs in a variety of situations are given.

**TITLE:** A Reidemeister-Schreier program

**AUTHOR:** George Havas

**CITATION:** *Proceedings of the Second International Conference on the Theory of Groups*, Lecture Notes in Mathematics **372** (1974) 347-356

**ABSTRACT:** The Reidemeister-Schreier method yields a presentation for a subgroup H of a group G when H is of finite index in G and G is finitely presented. This paper describes the implementation and application of a FORTRAN program which follows this method. The program has been used satisfactorily for subgroups of index up to several hundred.

**TITLE:** The two generator restricted Burnside group of exponent five

**AUTHORS:** George Havas, G.E. Wall and J.W. Wamsley

**CITATION:** Bulletin of the Australian Mathematical Society **10** (1974) 459-470 (open access)

**ABSTRACT:** The two generator restricted Burnside group of exponent five is shown to have order 5^{34} and class 12 by two independent methods. A consistent commutator power presentation for the group is given.

**TITLE:** Defining relations for the Held-Higman-Thompson simple group

**AUTHORS:** John J. Cannon and George Havas

**CITATION:** Bulletin of the Australian Mathematical Society **11** (1974) 43-46 (open access)

**ABSTRACT:** A set of defining relations for the Held-Higman-Thompson simple group of order 4 030 387 200 is given.

**TITLE:** Computational approaches to combinatorial group theory

**AUTHOR:** George Havas

**CITATION:** *PhD Thesis*, The University of Sydney (1974) 237 pages. [Scanned: 7MB]

**ABSTRACT:** Bulletin of the Australian Mathematical Society **11** (1974) 475-476 (open access)

**TITLE:** Some complexity problems in algebraic computations

**AUTHOR:** George Havas

**CITATION:** *Proceedings The Complexity of Computational Problem Solving*, University of Queensland Press (1976) 184-192.

**TITLE:** Computer aided determination of a Fibonacci group

**AUTHOR:** George Havas

**CITATION:** Bulletin of the Australian Mathematical Society **15** (1976) 297-305 (open access)

**ABSTRACT:** The Fibonacci group F(2, 7) has been known to be cyclic of order 29 for about five years. This was first established by computer coset enumerations which exhibit only the result, without supporting proofs. The working in a coset enumeration actually contains proofs of many relations that hold in the group. A hand proof that F(2, 7) is cyclic of order 29, based on the working in computer coset enumerations, is presented here.

**TITLE:** Collection

**AUTHORS:** George Havas and Tim Nicholson

**CITATION:** *Proceedings SYMSAC '76*, ACM Symposium on Symbolic and Algebraic Computation, ACM (1976) 9-14

**ABSTRACT:** Collection processes have been the basis of group investigations by many people, some using hand calculation, some machine calculation. We describe a collection process which is specially efficient in the context of nilpotent quotient algorithm programs. The principles underlying our collection process are applicable in general.

**TITLE:** A computer aided classification of certain groups of prime power order

**AUTHORS:** Judith A. Ascione, George Havas and C.R. Leedham-Green

**CITATION:** Bulletin of the Australian Mathematical Society **17** (1977) 257-274; Corrigendum: ibid. 317-319 (open access); Microfiche supplement: ibid. 320

**ABSTRACT:** A classification of two-generator 3-groups of second maximal class and low order is presented. All such groups with orders up to 3^{8} are described, and in some cases with orders up to 3^{10}. The classification is based on computer aided computations. A description of the computations and their results are presented, together with an indication of their significance

**TITLE:** Groups of exponent eight.

**AUTHORS:** Fritz J. Grunewald, George Havas, J.L. Mennicke and M.F. Newman

**CITATION:** Bulletin of the Australian Mathematical Society **20** (1979) 7-16 (open access)

**ABSTRACT:** This paper is a survey of the current state of knowledge on groups of exponent 8. It contains a report on a first stage of an attempt to answer the Burnside questions for these groups.

**TITLE:** Integer matrices and abelian groups

**AUTHORS:** George Havas and Leon S. Sterling

**CITATION:** *Symbolic and algebraic computation*, Lecture Notes in Computer Science **72** (1979) 431-451

**ABSTRACT:** Practical methods for computing equivalent forms of integer matrices are presented. Both heuristic and modular techniques are used to overcome integer overflow problems and have successfully handled matrices with hundreds of rows and columns. Applications to finding the structure of finitely presented abelian groups are described.

**TITLE:** The last of the Fibonacci groups.

**AUTHORS:** George Havas, J.S. Richardson and Leon S. Sterling

**CITATION:** Proc. Roy. Soc. Edinburgh 83A (1979) 199-203

**ABSTRACT:** All the Fibonacci groups in the family F(2,n) have been either fully identified or determined to be infinite, bar one, namely F(2,9). By using computer-aided techniques it is shown that F(2,9) has a quotient of order 152x5^{741}, and an explicit matrix representation for a quotient of order 152x5^{18} is given. This strongly suggests that F(2,9) is infinite, but no proof of such a claim is available.

**TITLE:** Application of computers to questions like those of Burnside

**AUTHORS:** George Havas and M.F. Newman

**CITATION:** *Burnside groups*, Lecture Notes in Mathematics **806** (1980) 211-230

**ABSTRACT:** Computers have been used in seeking answers to questions related to those about periodic groups asked by Burnside in his influential paper of 1902. A survey is given of results obtained with the aid of computers and a key program which manipulates presentations for groups of prime-power order is described.

**TITLE:** Groups of exponent eight

**AUTHORS:** Fritz J. Grunewald, George Havas, J.L. Mennicke and M.F. Newman

**CITATION:** *Burnside groups*, Lecture Notes in Mathematics **806** (1980) 49-188

**TITLE:** Commutators in groups expressed as products of powers

**AUTHOR:** George Havas

**CITATION:** Communications in Algebra **9** (1981) 115-129

**ABSTRACT:** It is well known that in a free group the simple commutator [Y, X] (alternatively called the first Engel word) can be expressed as a product of squares. Likewise the second Engel word [Y, X, X] can be expressed as a product of cubes. Results on groups of exponent four imply that the fifth Engel word [Y, X, X, X, X, X] can be expressed as a product of fourth powers, and explicit expressions have now been obtained with the assistance of a computer. The results and the computeraided technique are described.

**TITLE:** HYPERdisk, an access method for remote disk devices

**AUTHOR:** George Havas

**CITATION:** Australian Computer Journal 13 (1981) 64-65

**ABSTRACT:** A method currently under development for accessing IBM-compatible disk devices from IBM-compatible is described. The method is transparent to application programmer and utility user. It for data transfer over the Network Systems HYPERchannel which is the basis of the local computer network.

**TITLE:** The CSIRO HYPERchannel local computer network

**AUTHORS:** George Havas

**CITATION:** Proceedings Symposium on Local Area Networks (1982) 5pp.

**TITLE:** Groups of exponent five and class four

**AUTHORS:** George Havas and J.S. Richardson

**CITATION:** Communications in Algebra **11** (1983) 287-304

**ABSTRACT:** We investigate presentations for the freest two-generator group of exponent five and class four, and obtain a number of minimal presentations, including two which contain the least possible number of non-fifth-power relators. Our aims are threefold: firstly, to provide some partial evidence in favour of the finiteness of the Burnside group of exponent five on two generators; secondly, to examine a refinement of the well-known question concerning the existence of minimal presentations for finite p-groups; and thirdly, to illustrate several ways in which a number of available computer techniques can be combined to demonstrate the finiteness of a group with a given presentation.

**TITLE:** Minimal presentations for finite groups of prime-power order

**AUTHORS:** George Havas and M.F. Newman

**CITATION:** Communications in Algebra **11** (1983) 2267-2275

**ABSTRACT:** In his survey "Minimal presentations for finite groups" Wamsley asks: "Are there any four generator five relation finite groups?" A more precise formulation is: are there any finite groups which can be generated by 4 elements but not by 3 elements and which can be defined by 5 relations on a 4-element generating set ? The answer is yes. We describe four such groups and indicate how they were found. The groups are given in order of increasing size . The proof for the first is simply a (huge) coset enumeration. The other proofs are more interesting ; only one is spelt out.

**TITLE:** Two groups which act on cubic graphs

**AUTHORS:** George Havas and Edmund F. Robertson

**CITATION:** Computational group theory, Academic Press (1984) 65-68

**ABSTRACT:** N.L. Biggs studies automorphism groups which act on cubic graphs. In particular he describes two families of groups: 4+(a^l) and 5+(a^l). In the tables presented in his talk at the Symposium, Biggs indicated that the orders of 5+(a^16) and 4+(a^12) were unknown but that the groups have homomorphic images with orders 2^16.30.48 and 2^8.14.24. He asked for more information about these groups. We show that both these groups have considerably larger quotients than those quoted above and present here details of computations done at the Symposium which prove this. We expect that these computations may form the basis of a proof that both these groups are infinite.

**TITLE:** A Tietze transformation program

**AUTHORS:** George Havas, P.E. Kenne, J.S. Richardson and E.F. Robertson

**CITATION:** Computational group theory, Academic Press (1984) 69-73

Zbl. 569.20002 (G. Butler)

**ABSTRACT:** A Reidemeister-Schreier program which yields a presentation of a subgroup H of finite index in a finitely presented group G was described by Havas [Proceedings of the second international conference on the theory of groups, Lecture Notes in Math. 372 (1974) 347-356; MR 51#13002]. The program has two stages: first, Schreier generators and Reidemeister relators for H are computed; then the resulting presentation is simplified by eliminating redundant generators and by using a substring searching technique. The Tietze transformation program which we describe in this paper was originally designed to improve the simplification stage of that Reidemeister-Schreier program and now also forms part of the implementation of the modified Todd-Coxeter method [D. G. Arrell and Robertson, Computational group theory, Academic Press (1984) 27-32]. The program described here is written in a reasonably portable superset of FORTRAN 66, and was available at the symposium.

**TITLE:** Distinguishing eleven crossing knots

**AUTHORS:** George Havas and L. G. Kovács

**CITATION:** Computational group theory, Academic Press (1984) 367-373

**ABSTRACT:** Work has been done on the tabulation of knots since the last century. Ken Perko presented 552 distinct knots with eleven crossings and also provided a list of knot tabulations. In 1979 Richard Hartley drew our attention to Perko's work and to seven pairs of eleven crossing knots which Perko had not succeeded in distinguishing at that time. We indicate how we distinguished these pairs using group-theoretic calculations, not routinely applied by knot theorists. These knot pairs have now also been distinguished by Perko and by Thistlethwaite using more usual calculations.

**TITLE:** Local Computer Network Systems at CSIRO [In Japanese]

**AUTHORS:** George Havas and T. Tsukomoto

**CITATION:** FUJITSU 35 (1984) 107-115.

**TITLE:** CSIRONET - A national network for computer communication

**AUTHORS:** George Havas and P.J. Claringbold

**CITATION:** Proceedings ICCC'84 (1984) 56-63.

**TITLE:** CSIRONET research and development in Australia

**AUTHORS:** George Havas and P.J. Claringbold

**CITATION:** Proceedings World Computing Services Industry Conference IV (1984) 41-44.

**TITLE:** User experience with a very high speed local network

**AUTHOR:** George Havas

**CITATION:** Proceedings Lancon 84 (1984) 232-237.

**TITLE:** The connection between the Australian Bibliographic Network and CSIRONET

**AUTHORS:** W.S. Ford, George Havas and J.E. Paine

**CITATION:** Proceedings Second National ABN Conference (1985) 123-128.

**TITLE:** CSIRONET facilities for industry and government

**AUTHORS:** George Havas and P.J. Claringbold

**CITATION:** Proceedings First Pan Pacific Computer Conference (1985) 116, 1494-1515.

**TITLE:** The integration of diverse technologies by CSIRONET

**AUTHOR:** George Havas

**CITATION:** Proceedings VALA Third National Conference on Library Automation (1985) 37-41.

**TITLE:** Software product development and export

**AUTHORS:** George Havas and Howard Kadetz

**CITATION:** Proceedings ACC '86 (1986) 326-330