Finite Simple Groups

1. Finite Simple Groups Krista Lambroukos

2. 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

3. 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

4. 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”)

5. 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

6. Emile Mathieu (1835-1890) • Studied permutation groups during 1860’s • Discovered 5 finite simple groups that were not “Lie” type • First to do work with “sporadic” groups • M11, M12, M22, M23, M24

7. 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

8. Ludwig Sylow (1832-1918) • Developed theorems on powers of primes that divide the order of a group in 1870’s • Work was based on Lagrange’s Theorem • Provided future mathematicians tools to classifying more simple groups

9. 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 Sylow’s 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

11. Ferdinand Frobenius (1849-1917) • Changed the way of thinking in group theory to incorporate conjugacy classes • Elaborated on Sylow’s Theorems and introduced group characters and representation theory • Produced the irreducible characters for various groups in early 1900’s

12. 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) • “Burnside’s 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

14. 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

15. Richard Brauer (1901-1977) • Furthered the development of classifying finite simple groups using Frobenius’ group characters and character theory during the 1950’s

16. Walter Feit & John Thompson(1930-2004) (1932- ) • In 1963, the two proved Burnside’s 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

17. ZvonimirJankoMichio Suzuki(1932- ) (1926-1998) • During the 1960’s, both mathematicians classified other types of sporadic groups aside from Mathieu’s original five. • J1, J2, J3, J4 (J1 has order 175,650) • Suz or Sz has order 213 · 37 · 52· 7 · 11 · 13 = 448345497600

18. Daniel Gorenstein (1923-1992) • “Coach” • Guided the classification project and helped to organize the research pouring in • Declared the project complete in 1981

19. 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

20. Finite Simple Groups Song (of the 1960’s, 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 guessed:except of the cyclic ones, even the rest. Groups made up with permutes will produce more:For 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.

21. 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 defied:not 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 know:the 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 groups;and maybe we'd better go back to the loops.

22. References Doherty, F. (1997). A History of Finite Simple Groups. Retrieved November 27, 2010 from:http://math.ucdenver.edu/graduate/thesis/fdoherty.pdf 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 Gallian, J. (2010). Contemporary Abstract Algebra. Belmont, CA: Brooks/Cole Cengage Learning. O’Connor, 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. 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). ZvonimirJanko: outstanding Croatian mathematician. Retrieved November 27, 2010 from: http://www.croatianhistory.net/etf/janko/index.html.