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.

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.

4
In the Beginning

• All Zn where n is a prime or 1 are the only
  Abelian simple groups.
• Galois 1831 An for n gt 5
• Jordan 1870 Four families of matrix groups.
• A few more families and five sporadic groups are
  found between 1892-1905.

5
In the Beginning

• Jordans 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
1950s

• 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

8
1950s

• 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
1960s

• 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.

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  papers, publishd and unpublished.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.

13
The Proof

• Gaps discovered
• 1990s 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 gt 5
• Chevalley Groups
• Twisted Chevalley Groups
• Sporadic Groups
• Only five have order lt 1000
• Only 56 have order lt 1,000,000

15
The Groups

• 18 infinite families
• 26 sporadic groups
• Only five non-Abelian simple groups have have
  order lt 1000
• Only 56 have order lt 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.

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.

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

• Lets look at the numbers 1-200
• First determined by Hölder 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

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 Cauchys 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 ge, 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 lt q lt r.
• Show that groups of the following are not simple,
  using theorems 1 and 2
• a) 144
• b) 170
• c) 228