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]

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

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Mathieu group M<sub>22</sub> Sporadic simple group

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Mathieu group M<sub>23</sub> Sporadic simple group

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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> Sporadic simple group

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> Sporadic simple group

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.
  14. ↑ "ATLAS of Finite Group Representations - V3".
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