# Mathematicians

## Mathematicians involved in the Classification

Here are some of the mathematicians involved in my book *Symmetry and the Monster*.

### Late 18th century to mid 20th century

#### Joseph Louis Lagrange (1736–1813)

Born Guiseppe Lodovico Langrangia in northern Italy, he became professor in Berlin for more than 20 years, before taking up a position in Paris. He was one of the great mathematicians, working on many different aspects of mathematics: the three body problem; differential equations; number theory; probability; mechanics; and the stability of the solar system. In particular he published an influential paper (Reflections on the Algebraic Solution of Equations) in 1770. This paper inspired the work of many others, including Galois. For biographical information see the St Andrews website, and Wikipedia.

#### Évariste Galois (1811–32)

Galois died in 1832 at the age of twenty. He was fatally wounded in a duel, but the night before the duel he wrote a long letter explaining his mathematical ideas. Among other things he studied the question of when an algebraic equation has solutions that can be expressed in terms of radicals (meaning square roots, cube roots, and so on). His method involved treating the solutions as objects that could be permuted among one another. The group of allowable permutations — the Galois group of the equation — reveals immediately whether the solutions can be expressed in terms of radicals, without knowing a single solution. Galois’ ideas were published in 1846, and have been extremely influential, leading to what is now known as Galois theory. For biographical information see the St Andrews website, and Wikipedia.

#### Augustin-Louis Cauchy (1789–1857)

Cauchy used clear and rigorous methods in studying calculus, and wrote several influential books on the topic. He also had wide-ranging interests and played a role in the early history of group theory. He proved a theorem showing that if the size of a group is divisible by a prime number p, then it has a subgroup of size p. For biographical information see the St Andrews website, and Wikipedia.

#### Camille Jordan (1838–1922)

Jordan’s 1870 Treatise on permutations and algebraic equations clarified and expanded the new subject of group theory, particularly in connection with Galois’s work. For biographical information see the St Andrews website, and Wikipedia.

#### Sophus Lie (1842–1899)

Lie was born in Oslo in 1842 (though at that time the city was called Christiania). He was a larger than life character who developed new methods for studying solutions to differential equations (equations involving rates of change). In this context he introduced groups in which each operation could be gradually modified — they are now known as Lie groups. Lie took up a chair in Germany in 1886, but returned to a chair in Norway a few months before his death in February 1899. For biographical information see the St Andrews website, and Wikipedia.

#### Wilhelm Killing (1847–1923)

Killing discovered Lie algebras independently of Lie’s work. He then went on to classify them, and from this classification the table of most finite ’symmetry atoms’ was created. For biographical information see the St Andrews website, and Wikipedia.

#### Élie Cartan (1869–1951)

In his PhD thesis, Cartan revised Killing’s proofs of the classification of Lie algebras. He then went on to make significant contributions to differential equations and geometry. More details, click here. For biographical information see the St Andrews website, and Wikipedia.

#### William Burnside (1852–1927)

Burnside wrote the first book on group theory in English, published in 1897, and developed the subject from the modern abstract point of view. In 1904 he proved that the size of any finite simple group that is non-cyclic must be divisible by at least three different prime numbers. For biographical information see the St Andrews website, and Wikipedia.

#### Leonard Eugene Dickson (1874–1954)

In 1901 he published a book showing how to obtain finite versions for most families of Lie groups. This was the start of the ‘periodic table’ of finite simple groups. For biographical information see the St Andrews website, and Wikipedia.

#### Richard Brauer (1901–77)

Brauer founded the ‘cross-section’ (i.e. involution centralizer) approach to classifying the finite simple groups. He also did leading work on the character theory of finite groups. For biographical information see the St Andrews website, and Wikipedia.

#### Claude Chevalley (1909–84)

Chevalley worked on group theory and ring theory and in 1955 published a paper showing how to obtain finite versions of Lie groups in all families. For biographical information see the St Andrews website, and Wikipedia.

#### Jacques Tits (1930–

Like Chevalley, Tits was also pursuing finite versions of Lie groups in all families, but in a geometric way rather than using Chevalley’s algebraic approach. It led him to create the theory of buildings (which are ‘multi-crystals’, not buildings in the usual sense), which he went on to develop in other important ways. In 2008, Tits was awarded the Abel Prize, jointly with John Thompson. For biographical information, see the St Andrews website, and Wikipedia.

#### Walter Feit (1930–2004)

Feit was an expert on the character theory of finite groups, and collaborated with John Thompson to prove the celebrated theorem (the Feit-Thompson theorem) showing that a finite simple group that is not cyclic must have even size. For biographical information, see the St Andrews website, and Wikipedia.

#### John Thompson (1932–

Thompson’s early work led to his collaboration with Walter Feit on the great Feit-Thompson theorem (above). He went on to deal with the cross-section method of classifying finite simple groups, and was involved in studying the Monster and the new simple groups inside it, one of which is named after him. In 2008, Thompson was awarded the Abel Prize, jointly with Jacques Tits. For biographical information see the St Andrews website, and Wikipedia.

#### Daniel Gorenstein (1923–1992)

Gorenstein was the first person to put forward a plan for classifying all the finite simple groups, and he was closely involved with steering this project forward. When it appeared complete, he started the project, in collaboration with Lyons and Solomon, of revising and rewriting it so that it would stand the scrutiny of future generations. For biographical information see the St Andrews website, and Wikipedia.

### The Classification and Discovery of the Sporadic Groups

A great many mathematicians were involved in the Classification project, but only a few are mentioned in the book, and the same is true here. No disrespect is intended to those who are missing—only people whose work appears in the book are mentioned here, and the book is not a complete history of the Classification. For that one needs to read the books by Gorenstein, and by Gorenstein, Lyons and Solomon. Here the main topic is the discovery of the sporadic groups.

#### Émile Mathieu (1835–90)

A French mathematical physicist who, as a student, studied permutation groups that are multiply transitive. His results yielded five simple groups that are not of ‘Lie type’. These are the Mathieu groups M11, M12, M22, M23 and M24. For biographical information see the St Andrews website, and Wikipedia.

#### Ernst Witt (1911–91)

Witt created the Witt design on 24 symbols. It gives a simple way of understanding the Mathieu groups, and proves their existence. For biographical information see the St Andrews website, and Wikipedia.

#### John Leech (1926–92)

Leech discovered the Leech Lattice in 24 dimensions by using the Witt design, and started studying its symmetry group. For biographical information see the St Andrews website, and Wikipedia.

#### John Conway (1937–

Conway studied the symmetries of the Leech Lattice from which he produced three new finite simple groups, along with others that had already been found by other methods. He later worked on the Monster and its moonshine connections. For biographical information see the St Andrews website, and Wikipedia.

#### Zvonimir Janko (1932–

In 1966, Janko published the first new exception since Mathieu’s groups a century earlier. It is now known as J1. He went on to discover three more: J2, J3 and J4. All Janko’s sporadic groups were discovered by the cross-section (involution centralizer) method. For more information see Wikipedia.

#### Michio Suzuki (1926–98)

Suzuki made important contributions to the classification project in the early days and proved a version of the Feit-Thompson theorem in an important special case. In the early 1960s he also discovered a new family of finite simple groups that subsequently turned out to be groups of Lie type. Later in the 1960s he discovered a sporadic group that bears his name. For biographical information, see Wikipedia, and the website from the University of Illinois where he worked.

#### Bernd Fischer (1936–

Fischer discovered several sporadic groups, three of which are known by his name. These are the Fischer groups Fi22, Fi23 and Fi24 (the last one is not simple but contains a large simple subgroup). Fischer also discovered the Baby Monster, from which emerged the Monster. This in turn produced two new sporadic groups, which are named after those who did most of the work on them: Thompson in one case, and Harada and Norton in the other. For further information see Wikipedia.

#### Donald Livingstone (1924–2001)

Livingstone and Fischer together created the character table of the Monster, with Michael Thorne writing the computer programs that they needed for the calculations.

#### Marshall Hall (1910–90)

Hall constructed Janko’s group J2 as a group of permutations on 100 symbols. For biographical information see the St Andrews website, and Wikipedia.

#### Graham Higman (1917–2008)

Higman worked on the construction of several sporadic groups that had been discovered by the cross-section (involution centralizer) method. For biographical information see the St Andrews website, or Wikipedia.

#### Donald Higman (1928–2006)

Higman, in collaboration with Charles Sims, adapted Hall’s construction of J2 to produce another group of permutations on 100 symbols that was a new finite simple group. This is the Higman-Sims group. For more information see Wikipedia.

#### Charles Sims (1938–

In addition to being a co-discoverer of the Higman-Sims group, Sims used permutation techniques to construct several other sporadic groups. In collaboration with Jeffrey Leon he constructed the Baby Monster. For biographical information see Wikipedia.

#### Robert Griess (1945–

Griess predicted the Monster independently of Fischer, using Fischer’s Baby Monster. He later constructed the Monster as the group of symmetries for an algebra in 196,884 dimensions. For further information see his homepage, or Wikipedia.

#### John McLaughlin (1923–2001)

McLaughlin created a new sporadic group (the McLaughlin group) as a group of permutations.

#### Arunas Rudvalis (1945–

Rudvalis predicted the existence of a new sporadic group (the Rudvalis group) as a group of permutations. It was later constructed by Conway and David Wales. For more information see Wikipedia.

#### Dieter Held (1936–

Held discovered the sporadic group that bears his name. He used the cross-section (involution centralizer) method, and the group was later constructed by Graham Higman and John McKay. For more information see Wikipedia.

#### Michael O’Nan (1943–

O’Nan discovered the sporadic group that bears his name. He used the cross-section (involution centralizer) method, and the group was later constructed by Charles Sims.

#### Richard Lyons (1945–

Lyons discovered the sporadic group that bears his name. He used the cross-section (involution centralizer) method, and the group was later constructed by Charles Sims. Then with Ronald Solomon and Daniel Gorenstein he undertook the Revision of the Classification, a project that continues to this day. For more information see Wikipedia.

#### Koichiro Harada (1941–

Harada studied one of the two previously undiscovered simple groups that emerged as subgroups of the Monster. It is named after him and Simon Norton. For more information see Wikipedia.

#### Simon Norton (1952–

Norton calculated a large amount of information on the Harada-Norton group, and on the Monster itself. He also collaborated with Conway on the strange moonshine connections with the j‑function in number theory. For more information see Wikipedia.

#### John McKay (1939–

McKay made the first observation of a numerical coincidence between the Monster and the j‑function in number theory. He also made other intriguing observations, some of which have since been elucidated. For more information see Wikipedia.

#### Igor Frenkel (1952–

With James Lepowsky and Arne Meurman, he created the Moonshine module, connecting the Monster and the j‑function. For more information, see Wikipedia.

#### James Lepowsky (1944–

With James Lepowsky and Arne Meurman, he created the Moonshine module, connecting the Monster and the j‑function. For more information see Wikipedia.

#### Arne Meurman (1956–

With James Lepowsky and Arne Meurman, he created the Moonshine module, connecting the Monster and the j‑function. For more information see Wikipedia.

#### Richard Borcherds (1959–

Borcherds created a Monster Lie algebra that led him to a proof of the Conway-Norton conjectures for the Moonshine module, an achievement for which he was awarded the Fields Medal in 1998. For biographical information see the St Andrews website, and Wikipedia.

#### Michael Aschbacher (1944–

Aschbacher was the greatest contributor to the Classification program, apart from Thompson. He and Stephen Smith eventually filled in the missing part of the program, the quasi-thin case. For some biographical information, see Wikipedia.

#### Stephen Smith (1948–

In joint work, Smith and Aschbacher filled in a gap in the Classification by finally nailing the quasi-thin case. For biographical information see his homepage.

#### Ronald Solomon (1948–

Solomon, together with Richard Lyons and Daniel Gorenstein undertook the Revision of the Classification, a project that continues to this day. For biographical information see Wikipedia.