From the review by Ascher Wagner
Mathematical Reviews MR0562002

The lemma referred to in the title is the following very well-known result in the theory of permutation groups: Let $G$ be a permutation group on a set $\Omega$; then $\sum_{g \in G} | \text{fix}_\Omega (G)| = |\text{orb}_\Omega (G) | |G$. This result is noteworthy for the number of important applications it has in combinatorics and finite geometry.

The author first gives a numerical illustration of this result and then carefully explains two well-known proofs. He then proceeds to the real subject-matter of the paper: "Nowadays the result is frequently, but quite incorrectly, referred to as 'Burnside's Lemma' or 'Burnside's Theorem'." The misattribution appears to be due solely to the fact that this result appears in W Burnside's much-read book [Theory of groups of finite order, second edition, Cambridge Univ. Press, Cambridge, 1911]. Well, most basic results can be found in textbooks!

The author first finds this result, though with the rather unimportant restriction that G is transitive on
This paper is certainly a useful contribution to the history of 19th century group theory. The reviewer would also like to think that the author's example could infect other research mathematicians to be a little more scholarly in their attributions.

Dedicated with love and respect to my father for his eightieth birthday, 15 October 1989.

Evidence from published sources is used to show that Cauchy's group-theoretical work was all produced in a few months of intense activity starting in September 1845.

From the Introduction.

Had he lived so long A-L Cauchy would have been 200 years old in August this year [1989]. He and B H Neumann could have given us all very great pleasure by attending each other's birthday celebrations. In this note I propose to prove three assertions about a paper that Cauchy wrote when he was 56. The first is that he was 56 years old when he wrote it; the second, that, contrary to their publication dates, Bertrand's work preceded Cauchy's; and the third, that it was Bertrand's manuscript which triggered Cauchy's interest.

It is quite common to identify scientific articles by year of publication. This sometimes transposes priorities or mis-represents the facts by a year or two, but it rarely matters. And in exceptional cases the rule is broken - the famous Premier Mémoire, for example, which was submitted by Galois to the Paris Academy in January 1831 but first published by Liouville towards the end of 1846, is always thought of as being a product of that earlier time. The case that I propose to examine in some detail is another unusual one in that traditional dating places it too early. Coincidentally, it is related circumstantially, though not in any way directly, to the Premier Mémoire: for Galois' work and the 1845 papers by Cauchy are the two sources that introduced group theory to mathematics.

The paper in question Mémoire sur les arrangements ..., is one of Cauchy's major contributions. According to long established bibliographical convention the year of publication was 1844, but I believe that it was written in September and October 1845 at the same time as the long series of Comptes rendus articles. One might feel that this small discrepancy should be worth no more than a footnote in my long-planned, half-written larger essay on the development of group theory in the nineteenth century. The justification for a free-standing paper is that this apparently minor bibliographical point has disproportionate consequences for our understanding of Cauchy's impetus and of the relationship between Cauchy's work and that of Bertrand. Besides, there is a substantial credibility gap to be bridged because the 1844 dating has been accepted by many authoritative scholars. Thus G A Miller, Hans Wussing, B L van der Waerden, for example, refer to his work on group theory as stretching over the years 1844-46. That would suggest a much longer period for his preoccupation with the subject than the few months that I will show to have been the case. Of course it is a small mistake and one which need not worry us greatly were it not for two points. The first is this. Every time we read an item that takes the 1844 date seriously, as for example, those quoted above, or Josephine Burns [The foundation period in the history of group theory (1913)], or Amy Dalian [Les travaux de Cauchy sur les substitutions (1980)], we may be a little doubtful whether the author has properly read Cauchy's text because the evidence against 1844 is clear in the published record. The second and more serious point is that from time to time incorrect inferences have been drawn from the incorrect premiss. Lubos Novy, for example writes:

First Cauchy published an extensive treatise in the 3rd volume of the 'Exercices d'analyse et de physique mathématique' in 1844, and in the autumn of 1845 he decided to publish the main results again in the Comptes Rendus; however, he soon became dissatisfied with repeating his earlier results and he developed new methods and results.

Since the chronology is wrong Novy's suggestions as to Cauchy's state of mind are unlikely to be right. A more significant example is to be found in [Les travaux de Cauchy sur les substitutions (1980), 310] where Amy Dalian writes:

These articles by Cauchy on substitutions, and in particular the notes in the Comptes Rendus of the Academy, were practically ignored by other mathematicians until about 1800. Several contributions by J Bertrand and by J A Serret make no reference to them and incorporate none of the ideas developed by Cauchy.

This comment makes good sense in relation to Serret but it is quite inappropriate (though true) in relation to Bertrand because, as I have already indicated, Bertrand's work was written before Cauchy's.

The Obituary begins as follows.

William Ferrar, who was elected to the London Mathematical Society in November 1922 and served for ten years in various capacities on its Council, was born in Bristol on 21 October 1893 and died In Oxford on 22 January 1990, aged 96. Although most of his working life was spent in the service of Hertford College and the University of Oxford, through his textbooks and examining he contributed significantly to mathematics courses and syllabuses nationwide.

His father, George William Parsons Ferrar, who was described as a lamplighter on his marriage certificate, and his mother, whose maiden name was Maria Susannah Dale, were not well off, and sometimes found it hard to find the 1d to pay for his school lunch. Bill (as he was known to most colleagues in later years) was the eldest of three brothers. His schooldays were happy and successful and he proceeded by scholarships from his first school, Ashley Down, to Queen Elizabeth's Hospital at Brandon Hill, and from there to Bristol Grammar School. At the Grammar School he was taught by Jeff Westcott, who inspired in his pupil a great love of pure mathematics but failed to inspire a similar feeling towards applied mathematics. In December 1911 Ferrar competed for scholarships at Oxford, and was successful at Queen's College, but failed to win the Balliol award which he had considered more prestigious. At that time a scholarship not only carried substantial academic standing but also was an important financial resource, paying, in Ferrar's case, £80 of the £I150 per annum that he needed. (It is perhaps a surprise to many of us that even in those pre-war days the Local Authority made up the difference in the form of a grant of £70 per year from the Bristol Municipal Charities.) Once at Oxford, where he matriculated in 1912, Ferrar set his sights not just on First Class Honours in his first University Examinations (known as Honour Moderations or Mods) but on a better First than that of the Balliol scholar. Although the two of them were far ahead of the other candidates, the Balliol scholar again won. In those days, the next ambition for a clever undergraduate mathematician after success in Mods was the Junior Mathematical Scholarship at Oxford. Ferrar's tutor at Queen's (C H Thompson) was sufficiently impressed by his abilities and prospects to send him to Cambridge the following vacation. There he worked for six weeks with G N Watson, who introduced him to the study of infinite series and convergence which would later dominate his research work. Back in Oxford, he carried off the Junior Mathematical Scholarship in January 1914, beating the Balliol scholar into second place.

From the Introduction.

In the preface to the first edition (1897) of his book [Theory of groups of finite order] on the theory of finite groups, William Burnside wrote: 'The subject is one which has hitherto attracted but little attention in this country; it will afford me much satisfaction if, by means of this book, I shall succeed in arousing interest among English mathematicians in a branch of pure mathematics which becomes the more fascinating the more it is studied.' He returned to this point in his presidential address delivered to the London Mathematical Society on 12 November 1908. The published version [On the theory of groups of finite order] begins as follows:

It has been suggested to me that I should take advantage of the present occasion to give an account of the recent progress of the theory of groups of finite order. That very considerable advance has been made in the last twenty, and especially in the last ten years, is undoubtedly the case. That advance, however, has been in a great variety of directions, and it is probably too soon, as yet. to present its different parts in their proper proportion and perspective. It would not, I think, be possible to do at the present time for the theory of groups of finite order what was done so ably for the allied theory of algebraic numbers by Professor Hilbert in his report of 1897.

But a more serious objection to any attempt on my part to give, on the present occasion, an account of the recent advance in the theory is that such an account would certainly be uninteresting to a considerable number of my audience. It is undoubtedly the fact that the theory of groups of finite order has failed, so far, to arouse the interest of any but a very small number of English mathematicians; and this want of interest in England, as compared with the amount of attention devoted to the subject both on the Continent and in America, appears to me very remarkable. I propose to devote my address to a consideration of the marked difference in the amount of attention devoted to the subject here and elsewhere, and to some attempt to account for this difference.

Nearly a hundred years later the situation is very different. Almost all university mathematics courses include some group theory, and at many universities deeper parts of the subject are offered as advanced options. For well over thirty years the beginnings of the theory of groups have even been available in some school syllabuses. Whereas back in 1897 (and still in 1908) William Burnside was the only British mathematician contributing seriously to the theory of groups, now there must be well over fifty whose main research interests lie in the area (and at its peak in the 1960s and 1970 the number must have been higher). My purpose in this note is to sketch this development. It will be a very personal account, but if it draws attention to an area of the history of mathematics where interesting researches may be made then it will have succeeded.

Nowadays group theory has several major strands: finite group theory; representation and character theory; infinite group theory; theory of Lie groups and algebraic groups; computational group theory. In 1897 finite group theory was already a sophisticated subject; character theory had just appeared in some papers of Frobenius; infinite group theory was in its infancy; so was the theory of Lie groups; algebraic groups and computational group theory were half a century away in the future. If I concentrate on finite groups it is only to keep this account within reasonable bounds. Surveys of the development of other parts of group theory during the last hundred years would be of great interest, but must wait upon another occasion.

From the Introduction.

There is an old question in optics that has been called Alhazen's Problem: given a spherical mirror and points $A, B$ in space, how can a point $P$ on the mirror be found, where a ray of light is reflected from $A$ to $B$? Since $P$ must lie in the plane containing $A, B$ and the centre of the sphere, this is really a two-dimensional problem: given a circular mirror and points $A, B$ in its plane, find the point on the circle where a ray of light is reflected by the mirror from $A$ to $B$. It is this version of the problem that we shall discuss here. Dorrie [Heinrich Dorrie, A hundred great problems of elementary mathematics: their history and solution (1965)] refers to it as Alhazen's Billiard Problem for reasons which are not far to seek. The name Alhazen honours an Arab scholar Ibn al-Haytham who flourished 1000 years ago. The problem itself can be traced further back, at least to Ptolemy's Optics written some time around AD 150. A charming account, full of interesting historical and bibliographical pointers, has been published by John D Smith [John D Smith, The remarkable Ibn al-Haytham, Math. Gazette 76 (1992)].

For my father for his ninetieth birthday: 15 October 1999

This paper is devoted to a historical study of axioms for group theory. It begins with the emergence of groups in the work of Galois and Cauchy, treats two lines of development discernible in the latter half of the nineteenth century, and concludes with a note about some twentieth century ideas. One of those nineteenth century lines involved Cayley, Dyck and Burnside; the other involved Kronecker, Weber (very strongly), Hölder and Frobenius.

From the Introduction.

As is well known, my father has long had an interest in axiomatics, especially the axiomatics of group theory. My purpose in this essay is to trace the origins and development of currently familiar axiom systems for groups. A far more general study of the origins of the concept of abstract group has been undertaken by Wussing who, however, treats the axiomatics rather differently. George Abram Miller has also written on the subject but perhaps a little erratically. Statements like:

It should perhaps be noted in this connection that an abstract group is a set of distinct elements which obey the associative law when they are combined and is closed with respect to the unique solutions of linear equations in the special form $ax = b$. This special form appears already in the 'Rhind Mathematical Papyrus'.

are not easy to make sense of. On the other hand, his observations:

During recent years accurate definitions of an abstract group have been formulated. These are principally due to Frobenius and Weber. Such definitions are stated at the beginning of the second volume of Weber's 'Algebra'. It is to be hoped that practical uniformity in regard to the definition of an abstract group may be attained ... the term abstract group is frequently defined very inadequately, even in recent works, ...

and

Men like S Lie (1842-1899) and F Klein (1849-1925) continued to use the term group without defining it except that they assumed that the product of two elements of a given group is contained therein and sometimes they assumed also the existence of the inverse of every element within the group.

and

H Weber (1842-1913) published a set of postulates for abstract groups of finite order in the Mathematische Annalen, volume 20 (1882), page 302, which were extensively adopted by later writers in various countries.

can be substantially justified.

Teachers and students are often heard to refer to 'the four axioms of group theory'. The conventional reference is to Closure, Associativity, existence of an Identity element, and existence of Inverses. And the capitalised nouns in this last sentence are sometimes used as labels for them. Thus, for example, a well-known (and very successful) presentation of the theory defines a binary operation as 'a means of combining two elements' and defines (G, ), where G is a set and is a binary operation defined on G, to be a group if the following four conditions hold.

CLOSURE For all $g_{1}, g_{2} \in G, g_{1} \circ g_{2} \in G$.

IDENTITY There exists $e \in G$ such that for all $g \in G, g \circ e = e \circ g = g$.

INVERSES For each $g \in G$ there exists $g^{-1} \in G$ such that $g \circ g^{-1} = g^{-1} \circ g = e$.

ASSOCIATIVITY For all $g_{1}, g_{2}, g_{3} \in G, g_{1} \circ (g_{2} \circ g_{3} ) = (g_{1} \circ g_{2} ) \circ g_{3}$.

Although this definition is tried and tested and is successful pedagogically (as are many others like it), I myself do not feel comfortable with it. If a means of combining two elements of the set $G$, or a binary operation on the set $G$, is not a function $G \times G \rightarrow G$ then what is it? And if it is, then what is the point of the axiom of Closure? In the Inverses axiom, what is the element $e$? Of course it should be the same $e$ as was postulated in the Identity axiom. But in that statement $e$ is, technically, a bound variable and has no value outside the sentence. By analogy, if I use $G$ to stand for a group in one paper, I can still use it to stand for something quite different, such as the gravitational constant, in another. If we were to change $e$ to $f$ in the Identity axiom it would still be perfectly acceptable - indeed, its meaning would be quite unchanged - and yet the Inverse axiom would no longer make sense. Thus in fact Identity and Inverse need to be combined in such a way that the scope of the existentional quantifier $\exists e$ covers both. Well, this exemplifies the fact that the needs of pedagogy are not always easy to accommodate to the precision of mathematics - or vice-versa. That, however, is the subject of a different essay. For the purposes of this one we may take Closure to be the definition of a binary operation or to be a conventional textbook version of it, and we will take the scope of the quantifier $\exists e$ in the Identity axiom to include the Inverses axiom.

Group theory as we have it nowadays took a long time to evolve. It emerged around the middle of the nineteenth century from work of Galois published in 1846 (but written between 1829 and 1832) and work of Cauchy published the previous year. Cauchy's line of thinking can be traced still further back, to work of Ruffini published between 1799 and 1814. Ruffini's ideas have been analysed by various authors. Since he seems not to have had much direct influence on his successors, except a little tangentially through inspiring Cauchy to study substitutions in 1812 (this work was published in 1815), the present paper begins with Galois and Cauchy.

From the Introduction and Acknowledgements.

Issai Schur was born on 10 January 1875 and died on 10 January 1941, on his 66th birthday. His achievements as a mathematician. particularly as an algebraist, and the story of his life, including the terrible treatment he suffered in Nazi Germany from 1933 to 1939, are well known. They are treated in easily accessible literature: see, for example, the article by H Boemer in the Dictionary of Scientific Biography, Vol. 12 and the biography in the MacTutor History of Mathematics Archive compiled by J J O'Connor and E F Robertson on the University of St Andrews web-page: https://mathshistory.st-andrews.ac.uk. But the most comprehensive and perspicuous account was given by Alfred Brauer in the memorial address on Issai Schur which he delivered at the Humboldt University of Berlin on 8 November 1960 on the occasion of the 150th anniversary of that university. A slightly modified version of his lecture was printed in the first volume of Schur's Gesammelte Abhandlungen (Collected Works, 3 volumes edited by A Brauer and H Rohrbach, Springer-Verlag, 1973). Our first purpose here is, by kind permission of the publisher, to make Brauer's address available in English translation. We present this in Section 2; we also present a translation of the tribute published by the editors of Mathematische Zeitschrift in 1955.

The remainder of this article is conceived as a commentary on Alfred Brauer's address. We shall briefly recall the periods of Schur's life in Berlin, Bonn and Tel-Aviv. We are very grateful to Professors F Hirzebruch and W Purkert for sending us interesting material about Schur's appointment and academic work at Bonn. Our thanks are also due to Professor M Sonis for letting us have some notes that shed light on the sad last two years of Schur's life in Palestine. However, our main purpose is to publish here for the first time a number of documents, most of which belong to Schur's daughter, Frau Hilde Abelin-Schur who lives in Bern, Switzerland. We give a transcription and, where the material is originally in German, a translation. Schur's granddaughter Ms Susan Abelin of Zurich has kindly put at our disposal copies of a number of these documents, mostly letters from eminent mathematicians of that time. including A Brauer, C Caratheodory, A Fraenkel, F G Frobenius, Emmy Noether, E Steinitz and H Weyl. We are presenting here only a small selection of them and confine ourselves to material that refers to Schur's life rather than to mathematical problems. We hope the latter will be the subject of further research and will be published in due course. In any event, we wish to record our gratitude to Frau Abelin-Schur and to express our appreciation of the service she has rendered to the history of mathematics by allowing us to publish some of these documents in this book. We record our gratitude also to Ms Abelin for providing us with the copies from which we have worked. The one document that does not come from Switzerland is the testimonial for Alfred Brauer written by Schur himself. The original is in the possession of Mrs Hilde Brauer of Chapel Hill , North Carolina, widow of Alfred Brauer, and we thank her for permission to include it. Where appropriate, reference to these documents will be made in the text in the form [Weyl, 9 March 1939].

We are grateful also to the editors of this volume and to B H Neumann, who read a draft of this paper with a helpfully critical eye.

From the Introduction.

The so-called Second Mémoire of Évariste Galois, written in 1830 and first published in 1846, is notoriously difficult to understand. Already the title Des équations primitives qui sont solubles par radicaux requires considerable thought. For, the word 'primitive'. which has a standard meaning in the context of group theory now, had no such context or meaning when Galois used it.

I have two goals in the present paper: first, to give an account of the development of the concept of primitivity in finite group theory in the 19th century; secondly, to provide commentary on the first part of the Second Mémoire. These may appear to be two separate projects for which two separate papers would be appropriate. I combine them into one because, as the reader will see, they are in fact so closely related as to be almost inseparable. Hence my choice of an ambiguous title.

I also have two categories of reader in mind: historians and mathematicians. For the former, but also because one cannot discuss the history of the mathematics without having the details in front of one, I give expositions of some basic theory of equations and groups. For the latter I rehearse some of the known facts about Galois' writings.

Almost certainly the Second Mémoire was written in 1830. It was first published in 1846 by Liouville and there have been several re-publications since, culminating in the critical edition of 1962 by Robert Bourgne and J-P Azra. It is an unfinished first draft, nowhere near as much revised or as important as the Premier Mémoire. Nevertheless, where the Premier Mémoire describes what we now think of as Galois Theory, the Second Mémoire focusses heavily on groups and, as we shall see, has had, through the work of Camille Jordan and others, an important influence on group theory. It provides moreover a significant piece of evidence about the workings of the mind of an extraordinarily creative and intuitive young genius. Either one of these facts would be reason to make it worthy of intensive study; together they are compelling.

The Second Mémoire is in two parts. The first is about finite soluble equations and groups, the second about the groups we now call $AGL(2, p)$ and $PSL(2, p)$. The present paper is concerned only with the first part - I propose to write about the second part on some other occasion.

That first part contains Galois' wonderful insight (shared, as we shall see, by N H Abel) that a primitive soluble equation has prime-power degree. Equivalently:

a finite primitive soluble permutation group has prime-power degree.

His insight goes deeper than this-in his famous testamentary letter to Auguste Chevalier written on 29 May 1832, in the night before his fatal duel, he explains correctly that (in modem language) if the primitive soluble group has degree $p^{\nu}$ then it may be thought of as consisting of affine transformations of the form $\nu \mapsto \nu A + b$ of a $\nu$-dimensional vector space $V$ over the field $F_{p}$ of integers modulo $p$. (Here $A$ is an invertible $\nu \times \nu$ matrix over $F_{p}$ and $b \in V$.) Although his insight is correct, and the theorems are quite as important as he believed them to be, the arguments he gives in the Second Memoire are not easy to understand. Indeed, very probably they are wrong. To be certain that they are wrong, and cannot be corrected or completed, one would need to be sure what Galois meant when he described an equation as being primitive. And that, as has already been said, is the subject of this essay.

The idea of primitivity is fundamental in permutation group theory. My purpose is not only to seek to elucidate what Galois could have had in mind, but also to trace its development to its modern form, which had been reached by 1870, the year when Camille Jordan's great Traite des Substitutions et des Équations algébriques appeared. As we shall see, there are two concepts which might naturally be taken to be what Galois intended. These merge in an interesting way in Jordan's doctoral thesis of 1860-61. A few years later Jordan had established not only the modem notion but also the main reasons for its importance.

One of our difficulties amice lector will be to distinguish Galois' use of the word primitif (usually occurring in the form primitive as an adjective qualifying the feminine noun équation) from my use of the modern technical term 'primitive' as it is used in the theory of permutation groups. Throughout the text of this paper, but excluding the quotations, italics will be used, as above, for Galois' French word and roman type for the modem one. Another minor difficulty in quotations will be with such usages as 'Oeuvres', 'OEuvres', or 'de Évariste ' and 'd'Évariste' , or 'que une' and 'qu'une'. In all such cases I have chosen the line of least resistance and sacrificed consistency for faithfulness to the original.

Abstract

This talk, delivered as the keynote lecture to British Society for the History of Mathematics Research in Progress 2008, is about the excitement to be had from studying history of mathematics. It discusses some of the challenges that arise from the fact that mathematics is a seriously technical subject. They are illustrated with episodes from the history of the mathematical theory of symmetry. In particular, it focuses attention on the nineteenth century, where the mathematics requires degree-level training, while the history requires linguistic and scholarly sensitivities rarely provided by university mathematics courses.

Mathematicians and historians of mathematics

For the purposes of this lecture I'd like to begin by identifying three varieties of scholar:

historians of mathematics;
mathematician historians;
mathematicians.

By historians of mathematics I mean scholars who know and love a lot of mathematics - more than T C Mits (The Celebrated Man-in-the-street, see Lieber and Liebe) - but who are trained as historians. By mathematician historians I mean scholars who are trained and operate primarily as mathematicians but who have developed a real interest in the history of their subject. By mathematicians I mean colleagues whose whole focus is mathematics. Nevertheless, they will have an amateur interest in the history of their subject - as most do.

My own position is that of mathematician historian, but until now probably 80% mathematician, 20% historian. The work I have done in history is mainly on nineteenth-century (and some twentieth-century) algebra. My theme is the challenges of this area:

- mathematicians generally lack the necessary historical and linguistic training;
- historians generally lack the mathematical training to cope with material at this level.

I propose to illustrate the asymmetry of history in relation to mathematicians and historians with a number of examples.

Challenges for historians

When studying the history of modern algebra one faces a basic and perhaps surprising question before one can get properly started.

Example 1

What is a group? What is group theory?

My point here is that the position one takes on these two questions is a major determining factor on how, as a historian, one writes about groups and group theory. The answer is time-dependent.

For those working on the nineteenth-century origins of group theory one perfectly sensible position to take is that

          groups = groups of permutations (substitutions)

Permutation group theory is a significant part of modern group theory (see Mathematical Reviews for example). In fact both Galois with his 'groupes de substitutions' and Cauchy with his 'systèmes de substitutions conjuguées' introduced groups as subsets of the group of all permutations of a set of size $n$ such as $\{1, 2, ..., n\}$, closed under composition. As a consequence much early group theory - most of it in fact - developed as the study of groups of substitutions. The challenge for historians is not only to understand this subject area but also to appreciate its place within algebra.

Another perfectly sensible position is that

          groups = groups of symmetries of (geometrical) objects.

This is now the accepted view of what groups are primarily for - the measurement of symmetry. Since an entity is defined as much by its function as by what it actually is, this is a valid description of what groups are and what group theory is about. It emerged in the late nineteenth century through the work of Felix Klein and others. Insofar as symmetry groups are groups of permutations of the elements (such as points, lines, or members of a set with structure) that make up the object whose symmetry is to be measured they are groups of permutations. Conversely, at least in the finite case, for any group $G$ of permutations of a set $X$ there is a relational structure on $X$ of which $G$ is the group of symmetries. Thus groups of permutations and groups of symmetries are essentially the same, even if the focus of study may be a little different.

A third possible position is that

          groups = abstract groups (two meanings)

On the one hand groups are popularly introduced (and have been for about 50 years) as objects satisfying the four famous axioms

          closure (under a binary operation),
          associativity,
          existence of identity,
          existence of inverses.

For many people these four axioms define groups and group theory - and especially the concept of group. Myself - I'm not so sure. For one thing, I'd not have four axioms: for me closure is part of the definition of a binary operation, not really an axiom of group theory. For another, I'm not convinced of the pedagogic value of going from the abstract to the concrete, and for me permutation groups or symmetry groups come first. Still, the fact is that for many mathematicians and for many historians of mathematics nowadays it is the axioms that define the subject (see for example Wussing (1969). Famously, however, every permutation group is an abstract group in this sense insofar as it satisfies the axioms; and conversely, to every abstract group there is, by Cayley's Theorem, a permutation group isomorphic to it in the sense that there is a one–one correspondence between the two which respects the group operations. Thus the three notions of group we have had so far simply provide different views of the same concept.

In fact the term 'abstract group' had another meaning from about 1880 to about 1960. Namely, an abstract group was a group described by generators and relations.
...

Challenges for mathematicians

Perhaps the greatest difficulties for mathematicians doing history arise from their over-developed knowledge of their subject: we make what seem natural assumptions about

- what could be in the original,
- what might have been in the original,
- what we would have expected to see there,

instead of checking

- what actually is there.

I have in mind that if one really understands mathematics, something like Pythagoras' Theorem for example, then it is all too easy to believe not only that someone called Pythagoras discovered the theorem but also that he or she had the proof that is familiar to us and that we believe to be canonical. This of course is a bad example because there are so very many well-loved proofs of Pythagoras' Theorem. I can best explain my point by illustrating it with another much-loved theorem.

Example 3.

Should we write Lagrange's theorem or Lagrange's Theorem?

Compare

          Theorem (Lagrange).
          If $G$ is a finite group and $H$ is a subgroup of it then the order of $H$ divides the order of $G$.

with

          Lagrange's Theorem.
          If $G$ is a finite group and $H$ is a subgroup of it then the order of $H$ divides the order of $G$.

Which is correct? What is the issue? Indeed, what is the difference?

The difference, it seems to me, is that in each case the former attributes the theorem to Lagrange, the latter does not. The former indicates - even if it does not explicitly say so - that Lagrange discovered and proved this theorem. I have come across very many mathematicians who believe this. But it is simply not true. Lagrange's Theorem is a name; it is not an attribution. The word 'Theorem' here is part of that name; it is a proper noun, not a common noun, and is therefore capitalised.

When Évariste Galois died aged 20 in 1832, shot in a mysterious early-morning duel, he had already created mathematics which, in the context of its time, was of such extraordinary novelty that experienced academicians failed to understand it. After his main manuscripts were published by Liouville in 1846, however, his name was soon immortalised by its use in the terms 'Galois Theory' and 'Galois groups'.

This article, which has been written to celebrate the 200th anniversary of his birth, focuses on a study of his relation- ship with his groups: how Galois defined them; how he used them; what he knew about them; and his inventiveness. It is conceived as a contribution to the history of mathematics but with a mathematical readership primarily in mind.

From the Introduction

Évariste Galois (1811-1832), who died on 31 May 1832 after being shot in a mysterious early-morning duel the previous day, was described by one of his biographers as a 'Révolutionnaire et Géomètre' (Dalmas [André Dalmas, Évariste Galois, Révolutionnaire et Géomètre (1956)]). As a republican and a revolutionary he was passionate but not - so far as I read the evidence - a great success. He was, however, a géomètre révolutionnaire, a revolutionary mathematician. His great contributions to mathematics were the invention of Galois Theory and a theory of groups. He created groups as a tool for his study of the theory of equations. Having done so, he went further and began to study them as objects of interest in their own right, that is to say, he embarked on a theory of groups. Galois was not alone in this. Cauchy invented his version of groups, and instituted a theory of them, about 16 years later, in 1845. Although they had some points in common, the discoveries of Galois and of Cauchy occurred in quite different contexts and were almost certainly independent. It is the former that are to be the focus of this article, which is devoted to a detailed study of groups in Galois' writings, together with an assessment of the originality of his ideas about them.

From the Context

As a reminder, and for context and reference, here is a brief chronology of Galois' short and somewhat tormented life:

25 October 1811: Évariste Galois born in Bourg-la-Reine, a town (now a suburb) about 10 km south of the centre of Paris, the second of three children born to Nicolas-Gabriel Galois and his wife Adelaïde-Marie (née Demante).

6 October 1823: Entered the Collège Louis-le-Grand. His six-year stay there started well but ended badly.

August 1828: Failed to gain entrance to the École Polytechnique.

April 1829: Aged 17, had his first article (on continued fractions) published in Gergonne's Annales de Mathématiques.

25 May and 1 June 1829: Submitted, through Cauchy, a pair of articles containing algebraic research to the Académie des Sciences in Paris. Poinsot and Cauchy were nominated as referees. The manuscripts are now lost; René Taton published evidence that Galois probably withdrew them in January 1830.

2 July 1829: Suicide of Évariste's father Nicolas-Gabriel Galois.

July or August 1829: Second and final failure to gain entrance to the École Polytechnique.

November1829: Entered the École Préparatoire, as the École Normale (later, since 1845, the École Normale Supérieure) was briefly called at that time.

February 1830: Re-submitted his work on equations to the Académie des Sciences in competition for the Grand Prix de Mathématiques. His manuscript was lost by the academy. The prize was awarded jointly to Abel (posthumously) and Jacobi for their work on elliptic functions.

April-June 1830: Had three items published in Férussac's Bulletin. One 'Sur la théorie des nombres' was (and is) of great originality and importance. Wrote the unfinished draft 'Des équations primitives qui sont solubles par radicaux', now known as the Second Mémoire.

December 1830: Another item published in Gergonne's Annales.

4 January 1831: Official confirmation of his provisional expulsion from the École Préparatoire in December 1830.

17 January 1831: Submitted his 'Mémoire sur les conditions de résolubilité des équations par radicaux', now often known as the Premier Mémoire, to the Académie des Sciences. It was given to Lacroix and Poisson to be examined.

10 May 1831: Arrested for offensive political behaviour; acquitted on 15 June 1831.

4 July 1831: Poisson, on behalf of Lacroix and himself, reported back negatively on the 'Mémoire sur les conditions de résolubilité des équations par radicaux'.

14 July 1831: Arrested on the Pont-neuf during a Bastille Day republican demonstration. Held in the Sainte-Pélagie prison.

23 October 1831: Convicted of carrying firearms and wearing a banned uniform; sentenced to six months further imprisonment.

16 March 1832: Released from Sainte-Pélagie prison during an outbreak of cholera in Paris and sent to live in the 'maison de santé du Sieur Faultrier', a sort of safe house.

Late May 1832: Mysteriously engaged to duel. There is little evidence and much contradictory conjecture as to by whom and about what.

29 May 1832: Wrote his Lettre testamentaire addressed to his friend Auguste Chevalier and revised some of his manuscripts.

30-31 May 1832: Shot in an early-morning duel; died a day later in Paris.

Abstract

Before his death in 1832, Évariste Galois had already published some valuable mathematics. The manuscripts he left behind included a memoir that had been rejected by the Académie des Sciences (Paris) in 1831 but which changed the direction of algebra after it was published by Liouville in 1846, two other major works, and a morass of minor items. There have been many editions since then, culminating in the great 1962 Édition critique by Bourgne and Azra. Although both the 1846 edition by Liouville and the 1962 edition by Bourgne and Azra have been described as 'definitive', there is evidence that the process of convergence to a truly definitive edition is a long one that is not yet complete - if it ever can be. That evidence is what this note addresses.

Évariste Galois and his mathematical writings

Évariste Galois (1811-1832), who died on 31 May 1832, shot in a mysterious early-morning duel the previous day, was described by one of his biographers as a 'Révolutionnaire et Géomètre' [Dalmas, 1956]. As a republican and revolutionary he was passionate but not a great success. He was, however, a géomètre révolutionnaire, a revolutionary mathematician. After his so-called Premier Mémoire was published by Joseph Liouville in 1846 it changed the direction of algebra, transforming the theory of equations from its classical form into what is now known as Galois theory, a major branch of 'modern' or 'abstract' algebra, which is taught as an advanced option in many university undergraduate courses in pure mathematics. Famously, he spent the eve of the fatal duel organising and correcting some of his papers and writing a long letter, now known as the Lettre testamentaire, to his friend Auguste Chevalier. In it he summarised his work, announcing discoveries that go considerably beyond what he had got around to writing up. He also, in effect, appointed Chevalier as his literary executor, and it was Chevalier who published (at Galois' express request) the testamentary letter in September 1832, who took charge of the manuscripts that Galois left behind, copied many of them, and in 1843 gave them to Joseph Liouville, who, three years later, published an edition of the 'Oeuvres mathématiques d'Évariste Galois' [Liouville, 1846]. Some comments on the long silent period from 1832 to 1846, ended by the sudden explosion of interest in Galois' work that was sparked by its publication in 1846, may be found in [Neumann, 2011].

Introduction

Our colleague Jackie Stedall, who has died of cancer aged 64, became a historian of mathematics at the age of 50. Although her academic career lasted less than fourteen years, it was greatly influential. Her nine books, more than twenty articles, and major contributions to the on-line edition of the manuscripts of Thomas Harriot show exceptional breadth and depth of learning. They are cogently argued and written in a lucid and attractive style. The purpose of this article is to pay homage to a great scholar and to provide a contemporary account of her life and work for the permanent record.

Jackie's life

Jackie was born in Romford, Essex, to John and Irene Barton (née Stakes), the eldest of three sisters. Her father, John Barton, was a public health inspector and the family moved around the country with his employment. When she was still very young they were in Morley (near Leeds in West Yorkshire) for a few years, then in Carlisle (Cumbria), before settling in Walsall (just north of Birmingham) where she attended Queen Mary's High School for Girls. From there she won a place to read mathematics at Girton College in Cambridge. She earned a BA degree (later MA) from Cambridge University (1972), MSc in Statistics from the University of Kent at Canterbury (1973), Postgraduate Certificate in Mathematics Education from Bristol Polytechnic (1991), and PhD in History of Mathematics from the Open University (2000). She joined the University of Oxford in October 2000 as Clifford Norton Student (non-stipendiary) in the History of Science at The Queen's College where she became my close friend and colleague, and indeed, much loved by all she had anything to do with there.

In her own words this was her fifth career. Following her studies at Cambridge and Canterbury she became for three years (1973-1976) a research assistant in the Department of Mental Health at the University of Bristol. Her first article comes from that time. From 1977 to 1981 she was Overseas Programmes Administrator for the overseas development charity War on Want, a post in which she could indulge her great love of travel. In 1981 Jackie married Jonathan Stedall, a documentary film director, in St Enodoc's Church in Cornwall, one of their favourite places. They had two children, Thomas and Ellie, who were (in her words) the focus of her third career as full-time parent 1982-1989. She was a teacher of mathematics at Selwyn School, Gloucester, from 1993 to 1997 before embarking on her PhD studies 1997-2000 funded by an Open University Studentship.

Her academic career, which began in earnest with the Clifford Norton Studentship at Queen's, included ten years from 1997 as Open University part-time Tutor (later called Associate Lecturer) on courses in Mathematics and History of Mathematics.
...

From the Introduction.

In May 2014, soon after the Mathematical Association Conference in Nottingham, and soon after the passing of Roger Wheeler, a staunch member of the Mathematical Association, who had been my teacher at school, I had the idea that my presidential address in April 2016 might be entitled 'Inspiring Teachers'. The concept was a talk that might trace my mathematical experiences from grammar school sixth form, through my development in retirement as a contributor to masterclasses for the UK Mathematics Trust and the Royal Institution, learning from the students and from the inspiring teachers at whose masterclasses I assist, to the small understanding of Key Stage 2 Mathematics that I am gradually acquiring through an hour a week with some Year 6 students and their remarkable teachers in a local primary school.

A month or so later, when the organising committee was giving shape to Conference 2016, my words were taken as the title of the conference itself. It is deliberately ambiguous and it was gratifying to me to find this last April that its ambiguity had struck melodious chords with other speakers. My presidential address, however, was conceived as a lecture. Like all my lectures it was designed to be an oral presentation. It was not designed to be written down and published as an article. Please bear that in mind gentle reader, and judge accordingly. If you find something of value here I shall be delighted; if not, I shall not be surprised.

Abstract

William Burnside (1852-1927) was professor of mathematics at Royal Naval College, Greenwich from 1892 until 1917. The following is an impressionistic account of his contributions to pure (but applicable) mathematics. It is written for a non-specialist readership and explains why he counts as one of the great twentieth-century British mathematicians, even although he worked in a rare kind of isolation, producing internationally famous work in Group Theory, but unable to persuade his British contemporaries of the value and interest of research in the subject.

William Burnside and his Work

From the late nineteenth century on there have been a number of distinguished mathematicians (among them T A A Broadbent and L M Milne-Thompson) who worked at Royal Naval College, Greenwich (RNC). The most distinguished, however, in terms of impact on mathematics, and the one who is now best remembered for his original discoveries, was William Burnside (1852-1927).

Burnside served as professor at RNC from 1885 until 1919. His duties were focussed on teaching naval officers the basic mathematics needed for navigation, gunnery, and naval engineering. He taught a full load but had little administration imposed upon him, and although research was not part of his formal duties, he had plenty of time to think about mathematics beyond the needs of his teaching. For details of his life and work the reader is referred to the obituaries by Forsyth and the articles by Everett, Mann and Young, and Hawkins. Here is a brief chronology:

1852 William Burnside born (2 July) in London.

1858 Orphaned (that is, his father died).

Educated at Christ's Hospital School.

1871 Entered St John's College Cambridge: later migrated to Pembroke College.

1875 Fellow of Pembroke College (until 1886): spent ten years coaching rowing and mathematics.

1885 Appointed professor at Royal Naval College, Greenwich.

1886 Married.

1887 Start of his steady output of original discoveries in mathematics.

1893 Elected Fellow of the Royal Society; start of his commitment to the theory of groups. which is what he is now remembered for.

1899 Awarded De Morgan Medal by the London Mathematical Society.

1906 President of the London Mathematical Society (until 1908).

1919 Retired from Royal Naval College, Greenwich.

1927 Died (21 August) aged 75.

Burnside's career exhibits features that are unusual amongst highly original and productive mathematicians. His first article was not published until 1883 when he was over thirty years old. On his appointment to the chair of mathematics at RNC Greenwich in 1885 his output was still only three articles totalling a mere twelve printed pages. The reputation that earned him the professorship must have been based almost entirely on his work as a lecturer, mathematical coach, and examiner in Cambridge (though it would be nice to think - even although there is no evidence at all - that an appointing committee consisting of naval men would not hold a man's predilection for rowing against him). His fourth paper was published in 1887, the year after his marriage, and from then on he published an average of four items a year, a good output for a mathematician. Over his lifetime he wrote approximately 160 papers and two books, though the two editions of his book on group theory could perhaps be counted as separate items since the second edition contains much material that is not to be found in the first.

Much of his work in the early years at Greenwich was on applied mathematics. He wrote on Boltzmann's theory of gases, conceived as the statistical mechanics of huge numbers of molecules moving freely subject to collisions between them. His paper, although not uncontroversial, was generally well-received and stimulated useful discussion. Inspired perhaps by his observations of the river Cam as a rowing man, and probably by the observed behaviour of the tidal wave that travelled round the world after the explosion of Krakatoa in August 1883, he wrote two papers on hydrodynamics, the first on waves in deep water and the second on small waves essentially in shallow water. His studies in hydrodynamics and in electromagnetic theory were to lead him to questions in pure mathematics, in the theory of elliptic functions; he also wrote on a variety of technical problems in geometry and the theory of functions (higher calculus). It was this work totalling just over thirty papers that earned him election to Fellowship of the Royal Society in 1893. Late in his career, stimulated by military and naval needs arising from the Great War of 1914-1918, he turned his attention to probability and statistics. His thoughts in this area were summarised in the book that was published posthumously.

The year 1893 in which he was elected to the Royal Society was highly significant for Burnside's career also in another way. It was the year in which his first paper on the theory of groups was written. His contributions in this area have lasted far better than any of his other work. His name is attached to a number of concepts, theorems, and problems, and it is for his work in group theory that he is now remembered and revered. The second half of this paper will be an attempt to explain in a relatively nontechnical way what this subject is and what Burnside achieved in it.