is a Japanese

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

working on

finite group theory Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

. The

Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...

was Harada's first position in the United States in 1968. He graduated from

University of Tokyo , abbreviated as or UTokyo, is a public research university located in Bunkyō, Tokyo, Japan. Established in 1877, the university was the first Imperial University and is currently a Top Type university of the Top Global University Project by ...

in 1972.

Rutgers University Rutgers University (; RU), officially Rutgers, The State University of New Jersey, is a Public university, public land-grant research university consisting of four campuses in New Jersey. Chartered in 1766, Rutgers was originally called Queen's ...

was the scene from 1969 to 73 of his collaboration with

Daniel Gorenstein Daniel E. Gorenstein (January 1, 1923 – August 26, 1992) was an American mathematician. He earned his undergraduate and graduate degrees at Harvard University, where he earned his Ph.D. in 1950 under Oscar Zariski, introducing in his dissertati ...

on the classification challenge in finite groups. In 1971 he first taught at

Ohio State University The Ohio State University, commonly called Ohio State or OSU, is a public land-grant research university in Columbus, Ohio. A member of the University System of Ohio, it has been ranked by major institutional rankings among the best publ ...

, and in 1973 he was a visitor at

Cambridge University , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowledge. , established = , other_name = The Chancellor, Masters and Schola ...

where the Harada-Norton group was discovered. The

Gorenstein–Harada theorem In mathematical finite group theory, the Gorenstein–Harada theorem, proved by in a 464-page paper,

classifies finite simple groups of sectional 2-rank at most 4. In 1996 Ohio State held a Special Research Quarter on the

Monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order 246320597611213317192329314147 ...

and

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

s with ''Proceedings'' edited by Joseph Ferrar and Harada. In 2000

Mathematical Society of Japan The Mathematical Society of Japan (MSJ, ja, 日本数学会) is a learned society for mathematics in Japan. In 1877, the organization was established as the ''Tokyo Sugaku Kaisha'' and was the first academic society in Japan. It was re-organized ...

awarded Harada the Algebra Prize. After the

classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it ...

was announced, Harada proposed the following challenges to group theorists:Yasuhiko Tanaka (2003
Review: "Achievements and problems in the theory of groups"
in

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...

# Find natural

mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical pr ...

s realizing all

simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...

s as their

automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...

s. # Prove that there are only finitely many

sporadic simple group In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The ...

s. # Find the reason why the 26 sporadic simple groups exist. # Find a generalization of the Glauberman

Z* theorem In mathematics, George Glauberman's Z* theorem is stated as follows: Z* theorem: Let ''G'' be a finite group, with ''O''(''G'') being its maximal normal subgroup of odd order. If ''T'' is a Sylow 2-subgroup of ''G'' containing an involution no ...

. # Find an arithmetic to give the

Schur multiplier In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur multiplier \oper ...

s of finite simple groups. # Complete the theory of

modular representation Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field ''K'' of positive characteristic ''p'', necessarily a prime number. As well as h ...

s. # Classify the 2-groups that can be the

Sylow 2-subgroup In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed ...

s of finite simple groups. # Look for a completely new proof of the classification. # Classify finite simple groups having a strongly p-embedded subgroup. # Solve problems around the restricted Burnside problem.

Publications

* 1974: (with

) ''Finite simple groups whose 2-subgroups are generated by at least 4 elements'',

Memoirs of the American Mathematical Society ''Memoirs of the American Mathematical Society'' is a mathematical journal published in six volumes per year, totalling approximately 33 individually bound numbers, by the American Mathematical Society. It is intended to carry papers on new mathema ...

. * 1975: ''On the simple group F of order 2¹⁴ · 3⁶ · 5⁶ · 7 · 11 · 19''. Proc. Group Theory Conference in Park City, Utah, pp. 119–276. * 1989: ''Some elliptic curves arising from the Leech lattice'',

Journal of Algebra ''Journal of Algebra'' (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier. ''Journal of Algebra'' was founded by Graham Higman, who was its editor from 1964 to 1 ...

125: 289–310. * 1999: ''Monster''. Iwanami Publishing, (in Japanese; book on the

). * 2010: ''"Moonshine" of Finite Groups'',

European Mathematical Society The European Mathematical Society (EMS) is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians. The current ...

References

{{DEFAULTSORT:Harada, Koichiro 20th-century Japanese mathematicians 21st-century Japanese mathematicians Living people 1941 births