Finite Simple Groups

1. Finite Simple Groups Jessica Schaefer MAT 412C 14 April 2006

2. Simple Group A group G is simple if its only normal subgroups are {e} and itself. Example: Zp where p is a prime Facts are known about particular groups, but the information is rather �formless.� �Period� at end of last remark????�Period� at end of last remark????

3. Simple Groups They are the building blocks of all other groups. If G is a finite group and G1 is a normal subgroup of largest order, then G/G1 is a simple group. If G2 is a normal subgroup of largest order of G1, then G1/G2 is simple. These factors are called composition factors of G. You mention that simple groups are the building block of all other groups. HOW??? G2 normal subgroup of LARGEST ORDER of G1 -------------�Period� at end of last remarkYou mention that simple groups are the building block of all other groups. HOW??? G2 normal subgroup of LARGEST ORDER of G1 -------------�Period� at end of last remark

4. In the Beginning� All Zn where n is a prime or 1 are the only Abelian simple groups. Galois 1831: An for n > 5 Jordan 1870: Four families of matrix groups. A few more families and five sporadic groups are found between 1892-1905. ???Are you going to say more about each of these statements? Specifically will you say something about the matrix groups and the sporadic groups????Are you going to say more about each of these statements? Specifically will you say something about the matrix groups and the sporadic groups?

5. In the Beginning� Jordan�s groups SL(n, Zp)/Z(SL(n, Zp) n ? 2 and p ? 2 or 3 Certain groups of invertible linear transformations of a finite dimensional vector space over a finite field modulo the center of the group

6. In the Beginning� A simple group that does not fit into one of the infinite families of groups is called a sporadic group. The first ones found are known as the Mathieu groups

7. 1950�s A few ideas were developed to better understand and find simple groups. Centralizers of order 2 Normalizers of prime-power order subgroups Chevalley finds more simple families of groups ??? Again ---- will you be saying more about each of these ideas?????? Again ---- will you be saying more about each of these ideas???

8. 1950�s Chevalley groups are of Lie type. A Lie group is a group with the structure of a manifold. A manifold is a topological space that is locally Euclidian.

9. 1960�s Feit-Thompson Theorem A non-Abelian simple group must have even order. This had been conjectured around 1900. Thompson won the Fields Medal in 1970. Sadly, Feit was over 40. Will you be saying something about what the Fields Medal actually is?? Can you find the names of some other Fields Medal winners??Will you be saying something about what the Fields Medal actually is?? Can you find the names of some other Fields Medal winners??

10. The Quest The Feit-Thompson Theorem prompted the search to find all finite simple groups and prove no more exist. 1966-1975: 19 new sporadic groups discovered

11. The Monster 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 Found in 1980, it is a group of rotations in 196,883 dimensions.

12. The Proof In 1981 it was announced that the classification was complete. The proof ran over 10,000 pages in over 500 papers, publishd and unpublished. Over 100 mathematicians from the U.S., England, Germany, Australia, Canada, and Japan contributed to the proof. You have 10.000 and I think you mean 10,000!!!You have 10.000 and I think you mean 10,000!!!

13. The Proof Gaps discovered 1990�s: Michael Aschbacher and Stephen Smith begin work on the problem 2004: Aschbacher announces that the classification and proof are complete. The paper is about 1200 pages.

14. The Groups Zn where n is a prime or 1 An for n > 5 Chevalley Groups Twisted Chevalley Groups Sporadic Groups Only five have order < 1000 Only 56 have order < 1,000,000 Will you say something about what these groups are???Will you say something about what these groups are???

15. The Groups 18 infinite families 26 sporadic groups Only five non-Abelian simple groups have have order < 1000 Only 56 have order < 1,000,000 Steinberg, Suzuki, Ree

16. Tests for Nonsimplicity Theorem 1 Sylow Test for Nonsimplicity Let n be a positive integer that is not prime, and let p be a prime divisor of n. If 1 is the only divisor of n that is equal to 1 mod p, then there does not exist a simple group of order n. Again --- state it as a theorem.Again --- state it as a theorem.

17. Tests for Nonsimplicity Proof of Theorem 1 If n is a prime power, then a group of order n has a non-trivial center. If n is not a prime power then all Sylow p-subgroups are proper. Using the Third Sylow Theorem, all p-subgroups are unique, and therefore normal. Again -- I assume that you will be filling in the blanks here and that you are just listing the main ideas of the proof.Again -- I assume that you will be filling in the blanks here and that you are just listing the main ideas of the proof.

18. Tests for Nonsimplicity 100 = 22 * 52 Divisors: 1, 2, 4, 5, 10, 20, 25, 50, 100 1 is the only divisor of n that is equal to 1 mod 5. 84 = 22 * 3 * 7 Divisors: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 1 is the only divisor of n that is equal to 1 mod 7.

19. Tests for Nonsimplicity Let�s look at the numbers 1-200 First determined by H�lder in 1892 (range problem) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200

20. Tests for Nonsimplicity Our possible non-Abelian finite simple group candidates are: 12, 24, 30, 36, 48, 56, 60, 72, 80, 90, 96, 105, 108, 112, 120, 132, 144, 150, 160, 168, 180, 192 22 remain�

21. Tests for Nonsimplicity Theorem 2 2 * Odd Test An integer of the form 2n, where n is an odd number, greater than 1, is not the order of a simple group. Example: 46930 = 2 * 23565 It might be good to state this as a theorem.It might be good to state this as a theorem.

22. Tests for Nonsimplicity Proof of Theorem 2: |G | = 2n Tg = gx, for all x in G, g is an element in G g? Tg is an isomorphism form G to a permutaion group on the elements of G By Cauchy�s Theorem, G has an element of order 2, which will be our g.

23. Tests for Nonsimplicity When written in disjoint form, Tg has cycles of length 1 or 2. Cannot have any of length 1, otherwise g=e, and has order 1 Thus, Tg has n transpositions and Tg is an odd permutation. The set of even permutations in the image of G is a normal subgroup

24. Tests for Nonsimplicity 12, 24, 30, 36, 48, 56, 60, 72, 80, 90, 96, 105, 108, 112, 120, 132, 144, 150, 160, 168, 180, 192 19 possibilities remain�

25. Next Week More nonsimplicity tests The simplicity of A5 Homework due Wednesday, April 19th

26. Homework Show that there are no simple groups of order pqr, where p, q, r are primes and p < q < r. Show that groups of the following are not simple, using theorems 1 and 2: a) 144 b) 170 c) 228