# Charles Coffin Sims

### Quick Info

Elkhart, Indiana, USA

St Petersburg, Florida, USA

**Charlie Sims**was an American mathematician best known for his work in group theory and in particular for the discovery of one of the sporadic simple groups.

### Biography

**Charles Sims**was known to his friends and colleagues as Charlie. His parents were Ernest McPherson Sims (1883-1973) and Natalie Cornelia Coffin (1899-1987). Ernest McPherson Sims, born in Indianapolis, Indiana, on 1 April 1883, was the President and General Manager of a Metal Forming Company. He married Natalie Coffin on 6 March 1936 in Indianapolis, Marion, Indiana. Natalie Coffin, born in Indianapolis, Indiana, on 12 September 1899, was described as a housewife when Charles, the subject of this biography, was born. At this time the family were living at 509 Laurel Street, Elkhart, and Charles was born in Elkhart General Hospital. Two years later, on 12 September 1939, Charles' sister, Mary Jean Sims was born. At this time the family were living at 1901 Greenleaf Boulevard in Elkhart. At the time of the 1940 Census, the family are still living in Greenleaf Boulevard, but at this time Natalie is described as Secretary of her husband's Metal Forming Company.

Charles Sims attended both Elementary and High School in his home town of Elkhart, Indiana. After graduating from High School, he entered the University of Michigan where he first obtained an undergraduate degree then continued to undertake research for a Ph.D. at Harvard University advised by John Thompson. He looked at the number of isomorphism class of groups of order $p^{n}$ which Graham Higman had shown was $p^{An^{3}}$, where $A$, depending on $p$ and $n$, satisfies

$\large\frac{2}{27}\normalsize - o(n) ≤ A ≤ \large\frac{2}{15}\normalsize + o(n).$

In his thesis *Enumerating p-groups*Sims improved this result to show

$A = \large\frac{2}{27}\normalsize + O(n^{-\large\frac{1}{3}\normalsize} ).$

He was awarded a Ph.D. in 1963 and published a paper based on his thesis, and with the same title, in 1965. G Zappa, in a review of the paper states:-
The proof is very complicated and is based on several interesting propositions.While at Harvard, Sims met Annette, born 7 October 1941, when both were singing in the choir of Harvard-Epworth United Methodist Church on Massachusetts Avenue, Cambridge, Massachusetts. They later married in about 1969 and had two children, Mark Christopher Sims (born on 24 December 1971) and Amy Sims (later Amy Novack).

After the award of his doctorate, Sims taught for a short while at the Massachusetts Institute of Technology before being appointed to Rutgers University, New Brunswick, New Jersey, in 1965. Charles and Annette C Sims lived at first at 52 Terhune Road in Princeton, but later moved to Allenhurst, a small town on the Atlantic Coast near Asbury Park.

Sims' research soon turned to permutation groups and computational techniques in general. His second paper after the 1965 publication arising from his doctoral thesis, was

*On the group*(2, 3, 7; 9) (1968) in which he was able to use a computer implementation of the Todd-Coxeter coset enumeration algorithm to show a group to be infinite. He did this by proving the periodicity of the output of the algorithm. Sims' next paper was

*Computation of invariants in the theory of cyclotomic fields*(1966). In this paper he used an IBM 7094 computer to compute properties of the cyclotomic field of $p^{n+1}$th roots of unity over the rationals for all primes up to 4001. His next two papers were

*Graphs and finite permutation groups*I (1967) and

*Graphs and finite permutation groups*II (1968). In these papers he produces:-

... [a] relationship between transitive groups and graphs [which] gives new insight into known group-theoretic results and is applied to obtain new theorems about primitive groups which are not doubly transitive.In 1967 Sims made a major breakthrough when, working with Donald Higman, he discovered the previously unknown sporadic simple group now known as the Higman-Sims group. Using Donald Higman's theory of rank 3 groups, they constructed this group as a rank 3 extension of the Mathieu group $M_{22}$. The Higman-Sims group is a primitive permutation group of degree 100, has order 44 352 000 and has rank 3. They described the group in the paper

*A simple group of order*44 352 000 which appeared in

*Mathematische Zeitschrift*in 1968.

Sims made another major breakthrough which he announced to the conference

*Computational Problems in Abstract Algebra*held in Oxford, England, 29 August to 2 September 1967. The conference proceedings, edited by John Leech, was published (after a rather unfortunate delay) in 1970. In the paper corresponding to his talk, Sims wrote:-

One of the oldest problems in the theory of permutation groups is the determination of the primitive groups of a given degree. During a period of several decades a great deal of effort was spent on constructing the permutation groups of low degree. For degrees 2 through 11 lists of all permutation groups appeared. For degrees 12 through 15 the lists were limited to the transitive groups, while only the primitive groups of degrees 16 through 20 were determined. ... The basic assumption of this paper is that it would be useful to extend the determination of the primitive groups of low degree and that with recent advances in group theory and the availability of electronic computers for routine calculations it is feasible to carry out the determination as far as degree 30 and probably farther. Ideally one would wish to have an algorithm sufficiently mechanical and efficient to be carried out entirely on a computer. Such an algorithm does not yet exist. The procedure outlined in this paper combines the use of a computer with more conventional techniques. ... The paper concludes with a short description of a computer program for finding the order and some of the structure of the group generated by a given set of permutations and with a list of the 129 primitive groups of degree not exceeding 20.The computer program that Sims refers to in the final sentence is now known as an implementation of the Schreier-Sims algorithm, a linear time algorithm devised by Sims to find the order of a permutation group given generating permutations. The method allows other information about the permutation group to be found such as whether a given permutation is a member. The name of Otto Schreier is included since it is based on theory introduced by him. Sims writes in his 1970 article:-

A computer program of the type described has been written for the IBM 7040 at Rutgers. In its present form it can handle any group of degree 50 or less and can be of some use up to degree 127.Joachim Neubüser writes [4]:-

I think that Charles silently followed the good old saying that the proof of a pudding is in its eating and that similarly the proof of an algorithm is in its use and he did indeed prove the existence of three sporadic simple groups by constructing them as permutation groups using his 'Schreier-Sims' method.In fact Sims announced the existence and uniqueness of Lyons' group at the conference

*Finite groups '72*held at the University of Florida, Gainesville, Florida, 23-24 March 1972. At the conference

*Computational methods for representations of groups and algebras*at the University of Essen, Essen, 1-5 April 1997, Sims and George Havas gave a presentation for this Lyons finite simple sporadic group. They stated:-

We give a presentation of the Lyons simple group together with information on a complete computational proof that the presentation is correct. This fills a long-standing gap in the literature on the sporadic simple groups. This presentation is a basis for various matrix and permutation representations of the group.In 1977, in collaboration with Jeffrey Leon, Sims proved the existence and uniqueness of a simple group generated by {3, 4}-transpositions. This group, first conjectured to exist by Bernd Fischer in 1971, is generated by a class of involutions such that the product of any non-commuting pair has order 3 or 4.

Sims' outstanding research contributions were recognised in 1972 when he received the Rutgers University Board of Trustees Award for Excellence in Research.

Let us look now at some contributions that Sims made to the teaching and administration at Rutgers University. He was the chair of the new Faculty of Arts and Sciences Mathematics Department from 1982 until 1984 and was the Associate Provost during 1984-1988. His computer expertise was important to his service as Associate Provost for Computer Planning from 1984 to 1987 and later he served as Undergraduate Vice-Chair of the Mathematics Department from 1992 to 1999. In this last mentioned role we quote from the introduction he wrote to "News From the Undergraduate Program" in three Mathematics Department Newsletters. On 18 February 1998 Sims wrote [6]:-

This issue of the newsletter is one of the longest since publication began five years ago. A lot is happening in the undergraduate program. There are job opportunities for full- and part-time recitation instructors. We continue to teach more and more students. There are important developments in our courses and in the instructional strategies used by our teachers. Finally the Word Wide Web is having a dramatic impact on the entire educational process, not just in mathematics but across all disciplines.On 3 February 1999 Sims wrote [7]:-

This issue contains some good news and some bad news. For example, the department expects to receive a large grant from the National Science Foundation, there is a growing number of programs that involve undergraduates in mathematics research, and one of our tutorial assistant's had an important role in a Rutgers conference on teaching and learning. However, cheating on mathematics tests appears to be on the rise and the department does not have the resources needed to meet demand for certain kinds of mathematics courses. All of these topics deserve serious discussion.On 3 May 1999 Sims wrote [8]:-

Since I am stepping down as vice chair effective May 31, 1999, this is my last issue of the Newsletter. This issue summarizes the administrative changes taking place in the department. There are also descriptions of employment opportunities for undergraduates in the instructional program, reports of honours received by both students and faculty, and information about the effect of recent changes in our approach to teaching calculus.Sims also organised conferences and workshops at Rutgers. For example he organised a 'Workshop on Symbolic Software for Mathematical Research' at Rutgers University held 11-15 March 1991. His abstract, written shortly before the workshop took place, states:-

A workshop will be held to bring together leading mathematicians and experts in the development of symbolic software. The goal of the workshop will be to identify areas of mathematical research which would benefit from improvements in symbolic software, to discuss the form that new software should take, and to explore the possibility of setting up a multi-national development project to produce symbolic software needed for mathematical research. The workshop will focus primarily, but not exclusively, on research in such fields of algebra as number theory, group theory, algebraic geometry, and commutative algebra. The workshop is in part a response to recommendations in a recent report to the National Science Foundation urging increased activity in the field of symbolic computation.Sims wrote two books, the first being

*Abstract algebra: A computational approach*(1984). Boris Schein begins a review of the book as follows:-

This is a textbook for a one-year introductory course in abstract algebra with an emphasis on computation and algorithmic questions. However, it is not a book on applied algebra, and it is not a book on numerical linear algebra; the emphasis of the book is on exact computation. The goals of the book are to introduce students to the basic concepts and results of algebra, present the concept of an algorithm and discuss certain algebraic algorithms, show how computers can be used to solve algebraic problems and to provide a library of relevant computer programs, and to describe the APL computer language.The second book is the monograph

*Computation with finitely presented groups*(1994). We present a summary of Sims' Preface and his Introduction. Here is an extract from the Preface:-

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. In addition to a book like this one on computing with finitely presented groups, there would be books on computing with permutation groups, on computing with finite solvable groups, and on computing characters and modular representations of finite groups. Computational group theory was originated by individuals trained as group theorists. However, there has been a steadily increasing participation in the subject by computer scientists. There are two reasons for this phenomenon. First, group-theoretic algorithms, particularly ones related to permutation groups, were found to be useful in attacking the graph isomorphism problem, a central problem in theoretical computer science. Once computer scientists began looking at group-theoretic algorithms, it was natural for them to attempt to determine the complexity of these algorithms. Second, the techniques and data structures of computer science have proved valuable in improving existing group-theoretic algorithms and in developing new ones. This book is intended to be a graduate-level text. I have made a deliberate attempt to make the material accessible to students of both mathematics and computer science.Here is an extract from the Introduction:-

This book describes computational methods for studying subgroups and quotient groups of finitely presented groups. The procedures discussed belong to one of the oldest and most highly developed areas of computational group theory. In order to better understand the context for this material, it is useful to know something about computational group theory in general and its place within the area of symbolic computation. The mathematical uses of computers can be divided roughly into numeric and nonnumeric applications. Numeric computation involves primarily calculations in which real numbers are approximated by elements from a fixed set of rational numbers, called floating-point numbers. Such computation is usually associated with the mathematical discipline numerical analysis. Examples of numerical techniques are Simpson's rule for approximating definite integrals and Newton's method for approximating zeros of functions. One nonnumeric application of computers to mathematics is symbolic computation. Although it is impossible to give a precise definition, symbolic computation normally involves representing mathematical objects exactly and performing exact calculations with these representations. It includes efforts to automate many of the techniques taught to high school students and college undergraduates, such as the manipulation of polynomials and rational functions, differentiation and integration in closed form, and expansion in Taylor series.Joachim Neubüser wrote a review of

*Abstract algebra: A computational approach*and kindly sent us a copy. It can be read at THIS LINK.

Sims retired from Rutgers University in 2007 and, with his wife Annette, went to live in St Petersburg, Florida [2]:-

After retirement, Charles continued to help students learn mathematics by tutoring local children in St Petersburg. He was a devoted husband, brother, and father and active in church and church choirs throughout his life. [Charles and Annette] enjoyed their community, four dogs and cat, and spending time on Tampa Bay on their boat. Charles also collaborated with his sister Mary Jean on preserving family archives. He was a modest, kind, much-loved man who left lasting impressions on everyone he met or who benefited from his academic work.Sims and his wife lived at 2824 Coffee Pot Boulevard Northeast in St Petersburg, a house with a balcony, palm trees in the garden, and a wonderful view looking over the water of Coffee Pot Bayou.

In 2012 the American Mathematical Society introduced the Class of Fellows:-

The Fellows of the American Mathematical Society program recognizes members who have made outstanding contributions to the creation, exposition, advancement, communication, and utilization of mathematics.Charles C Sims, The State University of New Jersey New Brunswick, was elected to the Inaugural Class of Fellows.

Let us quote from a tribute written by Peter Cameron [1]:-

Sims was one of the most influential figures in computational group theory, but was much more besides. His name is attached to two sporadic simple groups; the one I know best, the Higman-Sims group, was found without any recourse to computation at all. It is a subgroup of index 2 in the automorphism group of a graph on 100 vertices, constructed from the 22-point Witt design. The graph had been constructed earlier by Dale Mesner ... Indeed, from my point of view, Higman and Sims were the two people who introduced graph theory into the study of permutation groups. I was lucky enough to be in on the ground floor, beginning my doctoral studies in 1968 (the year after the Higman-Sims group was found, though I wasn't yet in Oxford on that memorable occasion). ... I first met Sims in the early 1970s, when he came to a miniconference on permutation groups organised by Peter Neumann. One thing that I recall is that, at Peter's house, we were discussing perfect pitch; Charles pulled out his watch and asked me to put my ear to it and tell him the pitch of the tone. (As I recall, the mechanism caused it to vibrate at 360 hertz.)Sims died at the Marion and Bernard L Samson Nursing Center in St Petersburg, Florida. His funeral was held on Saturday, 11 November 2017, at the Anderson McQueen Funeral Home located at 2201 Dr Martin Luther King Street North, St Petersburg, Florida. It was followed by a committal service at The Church of the Beatitudes in St Petersburg.

### References (show)

- P Cameron, Charles Sims,
*Peter Cameron's Blog*(25 October 2017). https://cameroncounts.wordpress.com/2017/10/25/charles-sims/ - Charles Coffin Sims 1937-2017,
*Anderson McQueen Obituaries*. http://andersonmcqueen.com/tribute/details/89703/Charles-Sims/obituary.html#tribute-start - In Memoriam: Charles Sims (1937-2017),
*Department of Mathematics, Rutgers University*. https://www.math.rutgers.edu/for-alumni-friends/in-memoriam - J Neubüser, Charles Sims,
*Personal Communication*(31 October 2017). - Newsletter,
*Department of Mathematics, Rutgers University*(Spring 2007). http://sites.math.rutgers.edu/newsletter/newsletter2007.html - C C Sims, News From the Undergraduate Program,
*Mathematics Department, Rutgers University***5**(2) (18 February, 1998). - C C Sims, News From the Undergraduate Program,
*Mathematics Department, Rutgers University***6**(2) (3 February, 1999). - C C Sims, News From the Undergraduate Program,
*Mathematics Department, Rutgers University***6**(3) (3 May, 1999).

### Additional Resources (show)

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Written by J J O'Connor and E F Robertson

Last Update May 2018

Last Update May 2018