Marshall Hall Jr. (17 September 1910 – 4 July 1990) was an American

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, mathematical structure, structure, space, Mathematica ...

who made significant contributions to

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

and

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...

Career

Hall studied mathematics at

Yale University Yale University is a Private university, private research university in New Haven, Connecticut. Established in 1701 as the Collegiate School, it is the List of Colonial Colleges, third-oldest institution of higher education in the United Sta ...

, graduating in 1932. He studied for a year 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 ...

under a Henry Fellowship working with G. H. Hardy. He returned to Yale to take his Ph.D. in 1936 under the supervision of Øystein Ore. He worked in

Naval Intelligence Military intelligence is a military discipline that uses information collection and analysis approaches to provide guidance and direction to assist commanders in their decisions. This aim is achieved by providing an assessment of data from a ...

during

World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great power ...

, including six months in 1944 at

Bletchley Park Bletchley Park is an English country house and estate in Bletchley, Milton Keynes (Buckinghamshire) that became the principal centre of Allied code-breaking during the Second World War. The mansion was constructed during the years following ...

, the center of British wartime code breaking. In 1946 he took a position 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 pu ...

. In 1959 he moved to the

California Institute of Technology The California Institute of Technology (branded as Caltech or CIT)The university itself only spells its short form as "Caltech"; the institution considers other spellings such a"Cal Tech" and "CalTech" incorrect. The institute is also occasional ...

where, in 1973, he was named the first IBM Professor at Caltech, the first named chair in mathematics. After retiring from Caltech in 1981, he accepted a post at

Emory University Emory University is a private research university in Atlanta, Georgia. Founded in 1836 as "Emory College" by the Methodist Episcopal Church and named in honor of Methodist bishop John Emory, Emory is the second-oldest private institution of h ...

in 1985. Hall died in 1990 in

London London is the capital and List of urban areas in the United Kingdom, largest city of England and the United Kingdom, with a population of just under 9 million. It stands on the River Thames in south-east England at the head of a estuary dow ...

on his way to a conference to mark his 80th birthday.

Contributions

He wrote a number of papers of fundamental importance in group theory, including his solution of

Burnside's problem The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory and was inf ...

for groups of exponent 6, showing that a finitely generated group in which the order of every element divides 6 must be finite. His work in combinatorics includes an important paper of 1943 on

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that ...

s, which for many years was one of the most cited mathematics research papers.Mathematics Department of

br>Marshall Hall Jr.
via

Wayback Machine The Wayback Machine is a digital archive of the World Wide Web founded by the Internet Archive, a nonprofit based in San Francisco, California. Created in 1996 and launched to the public in 2001, it allows the user to go "back in time" and s ...

In this paper he constructed a family of non-Desarguesian planes which are known today as

Hall plane In mathematics, a Hall plane is a non-Desarguesian projective plane constructed by Marshall Hall Jr. (1943). There are examples of order ''p''2''n'' for every prime ''p'' and every positive integer ''n'' provided ''p''2''n'' > 4. Algebraic con ...

s. He also worked on block designs and

coding theory Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are stud ...

. His classic book on group theory was well received when it came out and is still useful today. His book ''Combinatorial Theory'' came out in a second edition in 1986, published by

John Wiley & Sons John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, i ...

. He proposed

Hall's conjecture In mathematics, Hall's conjecture is an open question, , on the differences between perfect squares and perfect cubes. It asserts that a perfect square ''y''2 and a perfect cube ''x''3 that are not equal must lie a substantial distance apart. This q ...

on the differences between

perfect squares In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The u ...

and

perfect cubes In arithmetic and algebra, the cube of a number is its third power, that is, the result of multiplying three instances of together. The cube of a number or any other mathematical expression is denoted by a superscript 3, for example or . T ...

, which remains an open problem as of 2015.

Publications

* 1943: "Projective Planes",

Transactions of the American Mathematical Society The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 ...

54(2): 229–77 * 1959: ''The Theory of Groups'', Macmillan ** Wilhelm Magnus (1960
Review: Marshall Hall, Jr. ''Theory of Groups''

Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. ...

66(3): 144–6. * 1964: (with James K. Senior) ''The Groups of Order 2ⁿ n ≤ 6)'', Macmillan ** Preface: "An exhaustive catalog of the 340 groups of order dividing 64 with detailed tables of defining relations, constants, and lattice presentations of each group in the notation the text defines. "Of enduring value to those interested in finite groups". * 1967: ''Combinatorial Theory'',

Blaisdell Blaisdell is a surname. Notable people with the surname include: *Alfred Blaisdell, politician from North Dakota *Daniel Blaisdell, politician from New Hampshire *Frances Blaisdell, musician from New Jersey *Frank Ellsworth Blaisdell (1862–1946), ...

Notes

References

* *

External links

* * {{DEFAULTSORT:Hall, Marshall Jr 1910 births 1990 deaths 20th-century American mathematicians Mathematicians from Missouri Scientists from St. Louis Algebraists Group theorists Emory University faculty Combinatorialists California Institute of Technology faculty Ohio State University faculty Yale Graduate School of Arts and Sciences alumni Yale College alumni