# Joachim Neubüser

### Quick Info

Belgard, Germany (now Białogard, Poland)

Aachen, Germany

**Joachim Neubüser**was one of the first to create a computer package to study discrete mathematics, in particular group theory. The system GAP, development of which bgan in 1985, was awarded the ACM/SIGSAM Richard Dimick Jenks Memorial Prize for Excellence in Software Engineering applied to Computer Algebra in July 2008.

### Biography

**Joachim Neubüser**was born in Belgard, Pomerania, a town of about 15,000 inhabitants about 25 km from the Baltic Sea coast. Today it is Białogard, in Poland, but when Joachim was born there it was part of Germany. Joachim was a friend and colleague of mine [EFR] whom I worked with for many years, so some details in the biography below come from things he told me about his life while some are from personal experiences working with him.

Joachim Neubüser was only six months old when, in January 1933, Adolf Hitler, the leader of the Nazi Party, became chancellor and head of the German government. Pomerania, part of Germany at this time, came under Nazi rule. Arguments between Germany and Poland over the region were used by Hitler as a reason to invade Poland in 1939 which marked the start of World War II. At this time Joachim was seven years old and had just begun has schooling. The war years were not easy and from 1943 onwards Pomerania became the target of Allied bombing. Many parts of Germany were finding conditions far worse, however, and families fled, particularly from Berlin, to Pomerania which was safer. At the beginning of 1945 Russian troops began attacking Pomerania which became a major battlefield. Families now sought safer places and the Neubüser family, like many others, moved west, settling in Oldenburg.

By May 1945 British and Canadian forces had advanced east as far as Oldenburg. German propaganda had painted the Allied troops as evil men who would murder and destroy, so Joachim and his family were terrified when they knew British and Canadian forces were approaching. A group of residents, including the Neubüser family, stood waiting in dread. As the soldiers reached them, the group pushed the twelve year old Joachim out, shaking with fear, to talk to the soldiers since he was the only one who knew a little English. He was overwhelmed by the kindness of the soldiers and there was still emotion in his voice when he told me [EFR] this story 45 years later.

Joachim Neubüser completed his schooling in Oldenburg in 1951 and later in that year he entered the University of Kiel to study mathematics. He worked for both a school teacher's certificate and for his doctorate in mathematics advised by Wolfgang Gaschütz.

After Gaschütz died, Joachim wrote to me [EFR] on 11 November 2016:-

[Wolfgang Gaschütz] has never worked on Computational Group Theory though he certainly caused my curiosity for the 'inside' of groups that lead me to my first attempts to look for a way of computing subgroup lattices.In 1957 Joachim Neubüser graduated from the University of Kiel with the Staatsexamen (the school teacher qualification), and his doctorate (Dr. rer. nat.) with his thesis

*Über homogene Gruppen*Ⓣ. He attended the International Congress of Mathematicians held at the University of Cambridge, England, in August 1958 and gave a short lecture on the results of his thesis.

The award of a British Council scholarship funded a postdoctoral year 1957-58 for Joachim at the University of Manchester. At this time Manchester was an exciting place for a young group theorist. Max Newman held the chair of Mathematics and worked on combinatorial topology but he had undertaken work on computers during World War II and had appointed Alan Turing to Manchester. He had also appointed to Manchester, among others, B H Neumann who was one of the leading group theorists. When Joachim Neubüser arrived in Manchester he was immediately introduced to four Ph.D. students of B H Neumann, namely Jim Wiegold, Gilbert Baumslag, Ian David Macdonald and Michael Frederick Newman. All four would become important figures the group theory with three of the four making important contributions to computational group theory. Jim Wiegold submitted his doctoral dissertation entitled

*Nilpotent products of groups with amalgamations*in 1958 and was awarded his Ph.D. Gilbert Baumslag was also awarded his Ph.D. in 1958; his thesis, was entitled

*Some aspects of groups with unique roots*. Both Ian D Macdonald and Mike Newman were near the beginning of their studies and both graduated with a Ph.D. in 1960, Ian Macdonald with the thesis

*On Groups with Conditions like Finiteness of Conjugacy Classes*and Mike Newman with the thesis

*On Groups All of Whose Proper Homomorphic Images are Nilpotent*.

The authors of [8] write:-

Joachim Neubüser arrived in Manchester with a list of hand-calculated Hasse diagrams of the subgroup lattices of all groups of order at most 31. It seems likely that he had already considered the idea to use computers for automating such calculations before arriving in England. Working in the mathematics department in Manchester clearly made this idea seem more feasible. He was encouraged in this view by Bernhard Neumann, who told him that although using computers in group theory might be unusual, it would be worth pursuing.After his postdoctoral year at Manchester, Joachim returned to Kiel in 1958 where he was appointed as an assistant to Wolfgang Gaschütz. Kiel had just purchased a Zuse Z22 computer and Joachim set about writing programs to compute subgroup lattices [8]:-

... Joachim Neubüser often spent nights and weekends at the Z22, programming the machine and watching it work. Programs were handed to the Z22 in the form of punched tape, and Joachim Neubüser was able to read the programs on these tapes fluently.This work led to his first two papers

*Untersuchungen des Untergruppenverbandes endlicher Gruppen auf einer programmgesteuerten electronischen Dualmaschine*Ⓣ (1960) and

*Bestimmung der Untergruppenverbände endlicher p-Gruppen auf einer programmgesteuerten elektronischen Dualmaschine*Ⓣ (1961). In the first of these he writes [13]:-

When examining group-theoretic problems, it is often useful to test the question using a larger sample of material. However, a more detailed discussion of special groups, if they are not of a well-known type such as abelian groups, can require a considerable amount of computation. It is therefore natural to try to use electronic calculators to study and discuss specific groups.Reviewing this paper, D H Lehmar writes:-

In group theory, there are hardly any systematic methods for investigating the structure of a given group; only a few calculation methods are used more frequently, such as that of residue class enumeration, for a group given by generators and defining relations, with the aim of obtaining a permutation representation for it. When discussing a group, one tends to make use of group-theoretic theorems that are applicable to the specific group, in order to avoid lengthy calculations as far as possible. The use of an electronic calculating machine, on the other hand, only makes sense if a systematic procedure is established which can be applied to any group or at least to a large classes of groups. To a limited extent (due to the memory capacity of the machine), changes in the calculation method based on already known partial results using group-theoretic theorems can be taken into account by branches of the program.

A description is given of a program designed to analyse the structure of finite groups with the aid of a binary digital computer instead of theorems. The group elements are represented by permutations on n marks. One or more words are used to store one permutation, using n subwords, each sufficiently large to store a number from 1 to n. Thus, for n = 19, 5 bits are required for each mark and so 7 marks are packed into each of 2 words and the remaining 5 marks are packed into a third word. (A 38 bit word is used.) The program generates the group from the given generators and proceeds to analyse the group by means of cyclic subgroups. It reports out (a) the order of the group, (b) the number of cyclic subgroups of prime power order, (c) the total number of subgroups, and (d) the number of classes of conjugate subgroups. Space and time restrictions of the machine, a Zuse 22, limit the program to groups of order about 300, without outside human assistance.The 1961 paper was reviewed by Brian Haselgrove who writes [9]:-

This paper describes a method for enumerating the subgroups of a group of order $p^{n}$ which has been programmed for the computer Zuse 22. It is found that a group of order $2^{7}$ with 514 subgroups and 334 classes of conjugate subgroups required 8 hours of computation, whereas another group of the same order with 386 subgroups and 224 classes required $5\large\frac{1}{2}\normalsize$ hours.In 1967 the first major conference on computation in abstract algebra was held in Oxford, England from 29 August to 2 September. By this time Joachim was head of perhaps the most active research group on computational group theory and he gave the major survey

*Investigations of groups on computers*, see [15]. His research group had six members, all of whom present at the meeting contributing several papers.

Now 1967 was an eventful year for Joachim Neubüser for in that year he completed work on his habilitation thesis

*Die Untergruppenverbände der Gruppen der Ordnung ≤ 100 mit Ausnahme der Ordnungen 64 und 96*Ⓣ. This led to his promotion at Kiel. Also in 1967 he married Freya Schwarck. Freya had studied for a doctorate at Kiel advised by Wolfgang Gaschütz and Friedrich Bachmann and had been awarded a Dr. rer. nat. in 1963 for her thesis

*Die Frattini-Algebra einer Lie-Algebra*Ⓣ. Freya was so kind to me when I visited them and she cooked a wonderful meal. She asked me what my favourite meal was and she cooked that for me on my next visit.

In 1969 Joachim was named chair of the Lehrstuhl D für Mathematik at the Rheinisch-Westfälische Technische Hochschule (RWTH Aachen University). The Lehrstuhl D für Mathematik was a new department, specially created to bring Joachim Neubüser to Aachen. He brought two colleagues from Kiel to help in founding the Department, namely Volkmar Felsch and Rolf Bülow. With their help Joachim began to quickly establish a major computational group theory team. He was now one of three major figures in computational group theory, the other two being Charles Sims and John Cannon. In the Preface of his 1994 book

*Computation with finitely presented groups*Sims wrote [19]:-

In 1970, John Cannon, Joachim Neubüser, and I considered the possibility of jointly producing a single book which would cover all of computational group theory. A draft table of contents was even produced, but the project was not completed. It is a measure of how far the subject has progressed in the past 20 years that it would now take at least four substantial books to cover the field, not including the necessary background material on group theory and the design and analysis of algorithms.Around 1971 Joachim Neubüser and John Cannon began a collaboration on a general purpose computational group theory system called the Aachen-Sydney Group System. This system has developed into MAGMA, a system widely used today for computations in algebra, number theory, algebraic geometry, and algebraic combinatorics. Joachim, however, left the project in the early 1980s mainly due to having a different vision about computer systems. You can see what these differences were by looking at the section "Some concerns" of Joachim's lecture

*A history of computational group theory*; see THIS LINK.

Let me [EFR] now say a little about how I collaborated with Joachim Neubüser. I was an undergraduate at the University of St Andrews, wrote a Ph.D. thesis on infinite dimensional matrix groups at the University of Warwick and was appointed to St Andrews in 1968. There I had two colleagues, Mike Beetham who had written a Todd-Coxeter coset enumeration program, and Colin Campbell who, advised by Willie Moser, had written an M.Sc. thesis on Nathan Mendelsohn's modification of Todd-Coxeter. I became interested in computational group theory and when I heard that Ian D Macdonald, who had produced one of the cleverest ideas in computational group theory, the $p$-Nilpotent-Quotient method, had moved from Australia to Stirling in Scotland, I invited him to lecture at St Andrews. We became good friends and, around the beginning of 1979 Ian told me that Joachim Neubüser had produced a counter-example to the class breadth conjecture. Ian said he wished he could find out the details, so I suggested that I invite Joachim to come to St Andrews. He came in the summer of 1979 and lectured to a small group including David Johnson who came from Nottingham with his Ph.D. student.

Joachim gave a series of excellent lectures on

*Counterexamples to the class-breadth conjecture*and while he was in St Andrews I asked him if he would be a main speaker if Colin Campbell and myself organised a bigger conference in 1981. Joachim agreed, provided we chose dates to fit in with the German school holidays. In addition, we invited Sean Tobin and Jim Wiegold to be main speakers and they agreed. With large numbers registering for the conference Groups St Andrews 1981, Joachim said we needed more than three main speakers so we invited Derek Robinson. These four gave five-lecture surveys which were published in the conference

*Proceedings*. Joachim's lecture series

*An elementary introduction to coset table methods in computational group theory*was expanded into a 45-page article in the

*Proceedings*. Let me explain here that Colin Campbell and myself were editors of the

*Proceedings*and made the (perhaps silly) decision to unify the style of the references in all the papers. When Joachim saw it he was furious we had changed his reference style and told me so in no uncertain terms. Then he said he'd got it off his chest, it was now all forgotten, no hard feelings. Unfortunately, we had to change Joachim's reference style in his article "History of computational group theory up to 1993" at THIS LINK so it would work in MacTutor. Joachim would have been furious.

The demand for the

*Proceedings*over the years was largely because of this article and it was so great that Cambridge University Press asked us to produce a second edition 25 years later. In this second edition we wrote in [3]:-

Even after 25 years the article by Joachim Neubüser remains the first source to which all three of us refer those who want to find out about the use of coset tables for studying groups. Our view is confirmed by the 14 Reference Citations from 1998 to 2005 which MathSciNet reveals for this article. ... One of Neubüser's aims in writing his survey was to provide a unified view on coset table methods in computational group theory. He addressed the way coset table concepts were developed, implemented and used.We note that from 1998 to 2022 there are now 25 Reference Citations to this article in MathSciNet.

A number of computer algebra systems were developed in Aachen by Joachim in collaboration with other colleagues. In 1981 CAS was announced which consisted of an interactive collection of programs designed to compute and work with character tables of finite groups. In 1982 the system Sogos became available based on Joachim Neubüser's algorithm to compute subgroup lattices. In 1989 SPAS was produced in Aachen containing versions of the modified Todd-Coxeter and Tietze Transformation programs which had been developed at St Andrews.

The system for which Joachim Neubüser is best known is GAP which began life around 1985. I was awarded an EPSRC grant 'A computer program to learn group theory techniques', 1984-87, to work with my Ph.D. student Kevin Rutherford on developing a computational group theory system. Some time in 1985 I suggested to Joachim that we apply to the European Communities (EC) (which became the EU in 1995), for funds to develop a computer algebra system. I told Joachim about the system I was developing with Kevin Rutherford and Joachim replied that he just begun developing a similar system called GAP. (Originally GAP stood for Groups And Programming, but was soon changed into Groups Algorithms Programming.) We decided to go ahead with an application to the EC for developing GAP. Our first application to the EC for a "twinning grant" was rejected on the grounds we were not asking for enough money. We then added Galway, Ireland (the EC seemed happy with 3 twins!) and asked for the grant 'Intelligent computer algebra systems' to support travel, research assistants, and equipment. This was awarded with myself as scientific coordinator which meant I did most of the administrative work but Joachim led the development of GAP. This was the first of four major EC grants, the other three being: 'Using computer algebra', 1993-94, involving co-operation between St Andrews, Aachen, Trento, and five Hungarian Universities; 'Computational Group Theory', 1993-96, involving co-operation between St Andrews, Aachen and nine other EC centres of computational algebra expertise; and 'Using computer algebra', 1994-96, involving co-operation between St Andrews, RWTH Aachen, Germany, Universita degli studi di Trento, Trento, Italy, Lorand Eotvos Budapest Hungary, Technical University of Budapest, Budapest, Hungary, Lajos Kossuth University, Debrecen, Hungary, and the University of Miskolc, Miscolc, Hungary.

I was the scientific co-ordinator for all four of these grants but I was greatly helped by Joachim in writing the reports which we would discuss in detail before I wrote them. To get an impression of the GAP development that came about from these grants, you can read a draft of the final report I wrote for the 'Computational Group Theory' one. I sent this draft to Joachim in March 1997 for his comments, see THIS LINK.

We organised Groups 1993 Galway / St Andrews. We wrote in the Preface:-

An invitation to Professor J Neubüser (Aachen) to arrange a workshop on Computational Group Theory and the use of GAP was taken up so enthusiastically by him that the workshop became effectively a fully-fledged parallel meeting throughout the second week, with over thirty hours of lectures by experts and with practical sessions organised by M Schönert (Aachen). These Proceedings contain an article by Professor Neubüser based on a lecture he gave in the first week of the conference.Joachim was keen to begin the GAP workshop at 8 a.m. each day but after a little gentle persuasion he agreed, somewhat reluctantly, to begin at 9 a.m.

On 31 July 1997 Joachim Neubüser retired and GAP headquarters formally moved from Aachen to St Andrews. He wrote on 30 June 1995:-

After careful consideration and discussion with various colleagues, I have come to the conclusion not to take the risk if my successor as chairman of Lehrstuhl D, who after all might get determined only shortly before my retirement, will be interested to further the development of GAP in a similar way as I have done. It seemed to be better and safer for GAP and the community of its users to hand the responsibility for GAP in time before my retirement to a group of colleagues who are both competent and willing to take over the work of the further development and maintenance of GAP.After he retired, Joachim, of course, continued to take a passionate interest in GAP development and was in frequent contact with me. GAP continued to be developed and it was awarded the ACM/SIGSAM Richard Dimick Jenks Memorial Prize for Excellence in Software Engineering applied to Computer Algebra in July 2008.

...

Let me finally state a point that is very important to me. The colleagues in St Andrew completely agree with me that also in their hands GAP will remain a system that can be obtained free of charge and with completely open source of kernel, library and data. Also the present support of the community of GAP users by GAP-forum and GAP-trouble will be maintained.

To read Joachim Neubüser's "History of computational group theory up to 1993" see THIS LINK.

In his 1993 lecture, Joachim Neubüser spoke about "Some concerns" looking at the positive and negative aspects of using a computer to solve algebra problems. For example, he wrote [16]:-

You can read Sylow's Theorem and its proof in Huppert's book in the library without even buying the book and then you can use Sylow's Theorem for the rest of your life free of charge, but ... for many computer algebra systems license fees have to be paid regularly for the total time of their use. In order to protect what you pay for, you do not get the source, but only an executable, i.e. a black box. You can press buttons and you get answers in the same way as you get the bright pictures from your television set but you cannot control how they were made in either case.For more of Joachim Neubüser's "Some concerns", see THIS LINK.

With this situation two of the most basic rules of conduct in mathematics are violated: In mathematics information is passed on free of charge and everything is laid open for checking. Not applying these rules to computer algebra systems that are made for mathematical research ... means moving in a most undesirable direction. Most important: Can we expect somebody to believe a result of a program that he is not allowed to see? Moreover: Do we really want to charge colleagues in Moldova several years of their salary for a computer algebra system?

In September 2021, Bettina Eick and Alexander Hulpke announced Joachim Neubüser's death to the GAP Forum [7]:-

It is with immense sadness we would like to inform you that Joachim Neubüser passed away on Saturday, September 25, after a long illness. He was the founder of the GAP System and also one of the main initiators of Computational Group Theory in its early days.Peter Cameron paid this tribute after the death of Joachim Neubüser [1]:-

...

Many modern group theoretic algorithms have been influenced by his work and the GAP System, of which he was the leader until his retirement 1997, is still one of the world leading computer algebra systems today.

As two of his doctoral students, both of us owe much of our education in group theory, and in algorithms, to him.

In Gratitude,

Bettina Eick and Alexander Hulpke.

Neubüser was much more than just the originator of GAP: a skilled computational group theorist himself, and an inspiring teacher. Indeed, when I wrote my book on permutation groups in the late 1990s, he wrote me a long account of the trials and mis-steps in the classification of transitive subgroups of low-degree symmetric groups, and permitted me to quote it in the book (which I did). But GAP, together with his students, make up his main legacy.

### References (show)

- P Cameron, Joachim Neubüser: Another sad loss (29 August 2021).

https://cameroncounts.wordpress.com/2021/09/29/joachim-neubuser/ - C M Campbell, Some examples using coset enumeration, in J Leech (ed.), Computational Problems in Abstract Algebra, Proc. Conf., Oxford, 1967 (Pergamon, Oxford, 1970)
- C M Campbell, G Havas and E F Robertson, Addendum to "An elementary introduction to coset table methods in computational group theory", in
*C M Campbell and E F Robertson (eds.), Groups-St Andrews 1981, St Andrews, 1981*(Second edition), London Math. Soc. Lecture Note Series**71**, (Cambridge University Press - C M Campbell and E F Robertson, Twenty-five years of Groups St Andrews conferences, in
*C M Campbell and E F Robertson (eds.), Groups-St Andrews 1981, St Andrews, 1981*(Second edition), London Math. Soc. Lecture Note Series**71**, (Cambridge University Press, Cambridge - C M Campbell and E F Robertson, The orders of certain Metacyclic Groups,
*Bull. London Math. Soc.***6**(3) (1974), 312-314. - H S M Coxeter, Review: Crystallographic groups of four-dimensional space, by Harold Brown, Rolf Bülow, Joachim Neubüser, Hans Wondratschek and Hans Zassenhaus,
*Bull. Amer. Math. Soc.***1**(5) (1979), 792-794. - B Eick and A Hulpke, The death of Joachim Neubüser,
*Gap Forum*(28 September 2021).

https://www.gap-system.org/ForumArchive2/2021/006322.html - B Eick, A Hulpke, Alice C Niemeyer, Joachim Neubüser (1932-2021), Obituary,
*Jahresbericht der Deutschen Mathematiker-Vereinigung***124**(2022), 147-155. - C B Haselgrove, Review: Bestimmung der Untergruppenverbände endlicher p-Gruppen auf einer programmgesteuerten elektronischen Dualmaschine, by J Neubüser,
*Mathematical Reviews*MR0134436**(24 #B489)**. - Joachim Neubüser,
*Lehrstuhl D für Mathematik, RWTH Aachen*.

http://www.math.rwth-aachen.de/~Joachim.Neubueser/ - Joachim Neubüser: Publications,
*Lehrstuhl D für Mathematik, RWTH Aachen*.

http://www.math.rwth-aachen.de/~Joachim.Neubueser/schrift.html - D H Lehmer, Review: Untersuchungen des Untergruppenverbandes endlicher Gruppen auf einer programmgesteuerten electronischen Dualmaschine (1960), by J Neubüser,
*Mathematical Reviews*MR0117939**(22 #8713).** - J Neubüser, Untersuchungen des Untergruppenverbandes endlicher Gruppen auf einer programmgesteuerten electronischen Dualmaschine,
*Numer. Math.***2**(1960), 280-292. - J Neubüser, Bestimmung der Untergruppenverbände endlicher p-Gruppen auf einer programmgesteuerten elektronischen Dualmaschine,
*Numer. Math.***3**(1961), 271-278. - J Neubüser, Investigations of groups on computers, in J Leech (ed.), Computational Problems in Abstract Algebra, Proc. Conf., Oxford, 1967 (Pergamon, Oxford, 1970)
- J Neubüser, An invitation to computational group theory, in
*C M Campbell, T C Hurley, E F Robertson, S J Tobin and J Ward (eds.), Groups '93 Galway/St Andrews Vol.***2**, London Mathematical Society Lecture Note Series**212**(Cambridge University Press, Cambridge - J Neubüser, The minster at Aachen,
*Lehrstuhl D für Mathematik, RWTH Aachen*(August 1999). - E F Robertson, Report on grant EC 'Computational Group Theory' No ERBCHRXCT930418.
- C C Sims,
*Computation with finitely presented groups*(Cambridge University Press, Cambridge, 1994).

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Written by J J O'Connor and E F Robertson

Last Update February 2023

Last Update February 2023