ATLAS of Finite Groups

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ATLAS of Finite Groups
ATLAS of Finite Groups cover.jpg
ATLAS of Finite Groups distinctive cherry red cardboard cover and spiral binding
Author
Subject Group theory
Publisher Oxford University Press
Publication date
December 1985
Pages294
ISBN 978-0-19-853199-9

The ATLAS of Finite Groups, often simply known as the ATLAS, is a group theory book by John Horton Conway, Robert Turner Curtis, Simon Phillips Norton, Richard Alan Parker and Robert Arnott Wilson (with computational assistance from J. G. Thackray), published in December 1985 by Oxford University Press and reprinted with corrections in 2003 ( ISBN   978-0-19-853199-9). [1] [2] The book codified and systematized mathematicians' knowledge about finite groups, including some discoveries that had only been known within Conway's group at Cambridge University. [3] Over the years since its publication, it has proved to be a landmark work of mathematical exposition. [1]

It lists basic information about 93 finite simple groups. The classification of finite simple groups indicates that any such group is either a member of an infinite family, such as the cyclic groups of prime order, or one of the 26 sporadic groups. The ATLAS covers all of the sporadic groups and the smaller examples of the infinite families. The authors said that their rule for choosing groups to include was to "think how far the reasonable person would go, and then go a step further." [4] [5] [6] The information provided is generally a group's order, Schur multiplier, outer automorphism group, various constructions (such as presentations), conjugacy classes of maximal subgroups, and, most importantly, character tables (including power maps on the conjugacy classes) of the group itself and bicyclic extensions given by stem extensions and automorphism groups. In certain cases (such as for the Chevalley groups ), the character table is not listed and only basic information is given.

The ATLAS is a recognizable large format book (sized 420 mm by 300 mm) with a cherry red cardboard cover and spiral binding. [7] (One later author described it as "appropriately oversized". [8] Another noted that his university library shelved it among the oversized geography books. [9] ) The cover lists the authors in alphabetical order by last name (each last name having exactly six letters), which was also the order in which the authors joined the project. [10] The abbreviations by which the authors refer to certain groups, which occasionally differ from those used by some other mathematicians, are known as "ATLAS notation". [11]

The book was reappraised in 1995 in the volume The Atlas of Finite Groups: Ten Years on. [12] It was the subject of an American Mathematical Society symposium at Princeton University in 2015, whose proceedings were published as Finite Simple Groups: Thirty Years of the Atlas and Beyond. [13]

The ATLAS is being continued in the form of an electronic database, the ATLAS of Finite Group Representations. [14]

Related Research Articles

<span class="mw-page-title-main">Monster group</span> Finite simple 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
   808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
   = 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
   ≈ 8×1053.

<span class="mw-page-title-main">Baby monster group</span> Simple finite group

In the area of modern algebra known as group theory, the baby monster groupB (or, more simply, the baby monster) is a sporadic simple group of order

<span class="mw-page-title-main">Sporadic group</span> Finite simple group type not classified as Lie, cyclic or alternating

In the mathematical classification of finite simple groups, there are 26 or 27 groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups.

<span class="mw-page-title-main">Conway group</span>

In the area of modern algebra known as group theory, the Conway groups are the three sporadic simple groups Co1, Co2 and Co3 along with the related finite group Co0 introduced by (Conway 1968, 1969).

<span class="mw-page-title-main">Mathieu group</span> Five sporadic simple groups

In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups M11, M12, M22, M23 and M24 introduced by Mathieu. They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They are the first sporadic groups to be discovered.

<span class="mw-page-title-main">Suzuki sporadic group</span>

In the area of modern algebra known as group theory, the Suzuki groupSuz or Sz is a sporadic simple group of order

<span class="mw-page-title-main">O'Nan group</span>

In the area of abstract algebra known as group theory, the O'Nan groupO'N or O'Nan–Sims group is a sporadic simple group of order

<span class="mw-page-title-main">Rudvalis group</span>

In the area of modern algebra known as group theory, the Rudvalis groupRu is a sporadic simple group of order

Robert Arnott Wilson is a retired mathematician in London, England, who is best known for his work on classifying the maximal subgroups of finite simple groups and for the work in the Monster group. He is also an accomplished violin, viola and piano player, having played as the principal viola in the Sinfonia of Birmingham. Due to a damaged finger, he now principally plays the kora.

<span class="mw-page-title-main">Robert Griess</span> American mathematician

Robert Louis Griess, Jr. is a mathematician working on finite simple groups and vertex algebras. He is currently the John Griggs Thompson Distinguished University Professor of mathematics at University of Michigan.

Janko group J<sub>2</sub> In mathematics, one of the sporadic simple groups

In the area of modern algebra known as group theory, the Janko groupJ2 or the Hall-Janko groupHJ is a sporadic simple group of order

Mathieu group <i>M</i><sub>11</sub>

In the area of modern algebra known as group theory, the Mathieu groupM11 is a sporadic simple group of order

Mathieu group M<sub>12</sub>

In the area of modern algebra known as group theory, the Mathieu groupM12 is a sporadic simple group of order

Mathieu group M<sub>22</sub>

In the area of modern algebra known as group theory, the Mathieu groupM22 is a sporadic simple group of order

Mathieu group M<sub>23</sub>

In the area of modern algebra known as group theory, the Mathieu groupM23 is a sporadic simple group of order

Mathieu group M<sub>24</sub> Sporadic set type of order 244823040

In the area of modern algebra known as group theory, the Mathieu groupM24 is a sporadic simple group of order

<span class="mw-page-title-main">McLaughlin sporadic group</span>

In the area of modern algebra known as group theory, the McLaughlin group McL is a sporadic simple group of order

Conway group Co<sub>2</sub>

In the area of modern algebra known as group theory, the Conway groupCo2 is a sporadic simple group of order

Conway group Co<sub>3</sub>

In the area of modern algebra known as group theory, the Conway group is a sporadic simple group of order

Conway group Co<sub>1</sub>

In the area of modern algebra known as group theory, the Conway groupCo1 is a sporadic simple group of order

References

  1. 1 2 Breuer, Thomas; Malle, Gunter; O'Brien, E. A. (2017). "Reliability and reproducibility of Atlas information". Finite Simple Groups: Thirty Years of the Atlas and Beyond. Contemporary Mathematics. Vol. 694. American Mathematical Society. pp. 21–32. arXiv: 1603.08650 . ISBN   978-1-470-43678-0.
  2. ↑ Curtis, Robert T. (2022). "John Horton Conway, 26 December 1937 — 11 April 2020". Biographical Memoirs of Fellows of the Royal Society. 72: 117–138. doi:10.1098/rsbm.2021.0034.
  3. ↑ Denton, Brian (October 1986). The Mathematical Gazette. 70 (453): 248. doi:10.1017/S0025557200139440.{{cite journal}}: CS1 maint: untitled periodical (link)
  4. ↑ ATLAS, p. vii.
  5. ↑ Steen, Lynn Arthur; et al. (December 1986). The American Mathematical Monthly. 93 (10): C85–C91. JSTOR   2322937.{{cite journal}}: CS1 maint: untitled periodical (link)
  6. ↑ Steinberg, R. (January 1987). Mathematics of Computation. 48 (177): 441. JSTOR   2007904.{{cite journal}}: CS1 maint: untitled periodical (link)
  7. ↑ Griess, R. L.; et al. (July 2021). "Selected Mathematical Reviews related to the work of John Horton Conway" (PDF). Bulletin of the American Mathematical Society. 58 (3): 443–456. doi:10.1090/bull/1744.
  8. ↑ Sin, Peter (2010). American Mathematical Monthly. 117 (7): 657–660. doi:10.4169/000298910x496804.{{cite journal}}: CS1 maint: untitled periodical (link)
  9. ↑ Zaldivar, Felipe (30 March 2010). "The Finite Simple Groups". MAA Reviews. Mathematical Association of America. Retrieved 29 April 2024.
  10. ↑ ATLAS, p. xxxii.
  11. ↑ Griess, R. L. (July 2021). "My Life and Times with the Sporadic Simple Groups" (PDF). Notices of the ICCM. 9 (1): 11–46. doi:10.4310/ICCM.2021.v9.n1.a2.
  12. ↑ The atlas of finite groups, ten years on. Cambridge, U.K. ; New York, NY, USA : Cambridge University Press. 1998. ISBN   978-0-521-57587-4 – via Internet Archive.
  13. ↑ Bhargava, Manjul; Guralnick, Robert; Hiss, Gerhard; Lux, Klaus; Pham, Huu Tiep (2017). Finite Simple Groups: Thirty Years of the Atlas and Beyond (PDF). Princeton NJ: American Mathematical Society. ISBN   9781470436780.{{cite book}}: CS1 maint: date and year (link)
  14. ↑ "ATLAS of Finite Group Representations - V3".
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