Finite Simple Groups

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Finite Simple Groups

• Krista Lambroukos

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What is a Finite Simple Group?

• Only normal subgroups are itself and the identity
• Building blocks
• Similar to primes in Number Theory and the
  periodic elements in Chemistry

Examples Abelian Simple Groups
where n 1 or is prime
Non-Abelian Simple Groups (harder to describe)
where n 5
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Classifying the Finite Simple Groups

• Each finite simple group is either
• - Lie type
• - cyclic groups of prime order
• - alternating groups
• - one of the 26 sporadic groups

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Classification Theorem

• Involved over 100 mathematicians in total from
• United States
• England
• Germany
• France
• Norway
• Japan
• Korea
• and others
• 1870 2004
• 10,000 pages in total (Enormous Theorem)

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Evariste Galois (1811-1832)

• Observations of polynomials of degree five
• Made the first distinction of finite simple
  groups in1832
• Died at the age of 20

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Emile Mathieu (1835-1890)

• Studied permutation groups during 1860s
• Discovered 5 finite simple groups that were not
  Lie type
• First to do work with sporadic groups
• M11, M12, M22, M23, M24

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Camille Jordan (1838-1922)

• Expanded the study of group theory in 1870 using
  Galois research
• Developed an organized system for understanding
  simple groups
• Classified the alternating and classical linear
  groups

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Ludwig Sylow (1832-1918)

• Developed theorems on powers of primes that
  divide the order of a group in 1870s
• Work was based on Lagranges Theorem
• Provided future mathematicians tools to
  classifying more simple groups

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Otto Holder (1859-1937)

• In 1892, published a paper proving all finite
  simple groups up to order 200 had been discovered
• Marked official start to the classification
  project
• Work relied on Sylows theorems

10
Frank Cole George Miller(1861-1926)
(1863-1951)

• Extended the list up to groups of order 2001 in
  1900

• Determined all simple groups up to order 660

• Further investigated Mathieu groups and
  classified them as sporadic
• since they did not produce infinitely many
  possibilities of other groups

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Ferdinand Frobenius (1849-1917)

• Changed the way of thinking in group theory to
  incorporate conjugacy classes
• Elaborated on Sylows Theorems and introduced
  group characters and representation theory
• Produced the irreducible characters for various
  groups in early 1900s

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William Burnside (1852-1927)

• Wrote the first book on group theory in English
  in 1897
• In 1911, observed that character theory could be
  used to prove nonabelian simple groups of
  odd order do not exist (not actually proven until
  50 years later)
• Burnsides Problem on finiteness of groups is
  still studied today

13
Philip Hall (1904-1982)

• Greatly inspired by Burnside, revived the study
  of group theory after World War I
• Formulated a systematic method for classifying
  groups of prime-power order in 1932, a
  fundamental source of modern group theory

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Claude Chevalley (1909-1984)

• In 1950, made a distinction within Lie-type
  groups, called Chevalley groups
• Showed how to obtain finite versions of Lie-type
  groups in all families
• Work was used to make a distinction between
  classical and sporadic groups

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Richard Brauer (1901-1977)

• Furthered the development of classifying finite
  simple groups using Frobenius group characters
  and character theory during the 1950s

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Walter Feit John Thompson(1930-2004)
(1932- )

• In 1963, the two proved Burnsides theory that
  every finite simple group has even order in a 255
  page journal, known as Feit-Thompson Theorem
• Thompson won the Fields Medal in 1970

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Zvonimir Janko Michio Suzuki(1932-
) (1926-1998)

• During the 1960s, both mathematicians
  classified other types of sporadic groups aside
  from Mathieus original five.
• J1, J2, J3, J4 (J1 has order 175,650)
• Suz or Sz has order 213 37 52 7 11 13
  448345497600

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Daniel Gorenstein (1923-1992)

• Coach
• Guided the classification project and helped to
  organize the research pouring in
• Declared the project complete in 1981

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Michael Aschbacher Stephen Smith(1944- )

• Closed the gaps within the classification
  project and took 7 years to correct errors within
  the proof
• Declared in 2004 that the project was complete
  and could now be regarded as a theorem

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Finite Simple Groups Song

• (of the 1960s, author unknown, Sung to the tune
  of "Sweet Betsy from Pike")
• What are the orders of all simple groups?I speak
  of the honest ones, not of the loops.It seems
  that old Burnside their orders has
  guessedexcept of the cyclic ones, even the
  rest.
• Groups made up with permutes will produce
  moreFor An is simple, if n exceedes 4.Then,
  there was Sir Matthew who came into
  viewexhibiting groups of an order quite new.
• Still others have come on the study this
  thing.Of Artin and Chevalley now shall
  sing.With matrices finite they made quite a
  list.The question is Could there be others
  they've missed?
• Suzuki and Ree then maintained it's the casethat
  these methods had not reached the end of the
  chase.They wrote down some matrices, just four
  by four,that made up a simple group. Why not
  make more?
• And then came up the opus of Thompson and
  Feitwhich shed on the problem remarkable
  light.A group, when the order won't factor by
  two,is cyclic or solvable. That's what's true.

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Song (continued)

• Suzuki and Ree had caused eyebrows to raise,but
  the theoreticians they just couldn't faze.Their
  groups were not new if you added a twist,you
  could get them from old ones with a flick of the
  wrist.
• Still, some hardy souls felt a thorn in their
  side.For the five groups of Mathieu all reason
  defiednot A_n, not twisted, and not
  Chevalley.They called them sporadic and filed
  them away.
• Are Mathieu groups creatures of heaven or
  hell?Zvonimir Janko determined to tell.He found
  out what nobody wanted to knowthe masters had
  missed 1 7 5 5 6 0.
• The floodgates were opened! New groups were the
  rage!(And twelve or more sprouded, to greet the
  new age.)By Janko and Conway and Fischer and
  Held,McLaughtin, Suzuki, and Higman, and Sims.
• No doubt you noted the last lines don't
  rhyme.Well, that is, quite simply, a sign of the
  time.There's chaos, not order, among simple
  groupsand maybe we'd better go back to the
  loops.

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References

• Doherty, F. (1997). A History of Finite Simple
  Groups. Retrieved November 27, 2010
  fromhttp//math.ucdenver.edu/graduate/thesis/fdoh
  erty.pdf
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• Elwes, R. (2006). An enormous theorem the
  classification of finite simple groups. Plus
  Magazine, Issue 41. University of Cambridge.
  Retrieved November 27, 2010 from
  http//plus.maths.org/issue41/features/elwes/index
  .html
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• Gallian, J. (2010). Contemporary Abstract
  Algebra. Belmont, CA Brooks/Cole Cengage
  Learning.
•  
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• OConnor, J. Robertson, E. (2010). The MacTutor
  History of Mathematics Archive. Retrieved
  November 27, 2010 from http//www-history.mcs.st-
  andrews.ac.uk/index.html.
•  
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• Solomon, R. (2001). A Brief Classification of the
  Finite Simple Groups. Retrieved November 27, 2010
  from http//www.ams.org/journals/bull/2001-38-03/
  S0273-0979-01-00909-0/S0273-0979-01-00909-0.pdf
•  
•  
• Zubrinic, D. (2007). Zvonimir Janko outstanding
  Croatian mathematician. Retrieved November 27,
  2010 from http//www.croatianhistory.net/etf/jank
  o/index.html.
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