Lecture 5.7: Finite simple groups

Transcription

1 Lecture 5.7: Finite simple groups Matthew Macauley Department of Mathematical Sciences Clemson University Math 410, Modern Algebra M. Macauley (Clemson) Lecture 5.7: Finite simple groups Math 410, Modern Algebra 1 / 8

2 Overview Definition A group G is simple if its only normal subgroups are G and e. Since all Sylow p-subgroups are conjugate, the following result is straightforward: Proposition (HW) A Sylow p-subgroup is normal in G if and only if it is the unique Sylow p-subgroup (that is, if n p = 1). The Sylow theorems are very useful for establishing statements like: There are no simple groups of order k (for some k). To do this, we usually just need to show that n p = 1 for some p dividing G. Since we established n 5 = 1 for our running example of a group of size M = 00 = 3 5, there are no simple groups of order 00. M. Macauley (Clemson) Lecture 5.7: Finite simple groups Math 410, Modern Algebra / 8

3 An easy example Tip When trying to show that n p = 1, it s usually more helpful to analyze the largest primes first. Proposition There are no simple groups of order 84. Proof Since G = 84 = 3 7, the Third Sylow Theorem tells us: n 7 divides 3 = 1 (so n 7 {1,, 3, 4, 6, 1}) n The only possibility is that n 7 = 1, so the Sylow 7-subgroup must be normal. Observe why it is beneficial to use the largest prime first: n 3 divides 7 = 8 and n Thus n 3 {1,, 4, 7, 14, 8}. n divides 3 7 = 1 and n 1. Thus n {1, 3, 7, 1}. M. Macauley (Clemson) Lecture 5.7: Finite simple groups Math 410, Modern Algebra 3 / 8

4 A harder example Proposition There are no simple groups of order 351. Proof Since G = 351 = , the Third Sylow Theorem tells us: n 13 divides 3 3 = 7 (so n 13 {1, 3, 9, 7}) n The only possibilies are n 13 = 1 or 7. A Sylow 13-subgroup P has order 13, and a Sylow 3-subgroup Q has order 3 3 = 7. Therefore, P Q = {e}. Suppose n 13 = 7. Every Sylow 13-subgroup contains 1 non-identity elements, and so G must contain 7 1 = 34 elements of order 13. This leaves = 7 elements in G not of order 13. Thus, G contains only one Sylow 3-subgroup (i.e., n 3 = 1) and so G cannot be simple. M. Macauley (Clemson) Lecture 5.7: Finite simple groups Math 410, Modern Algebra 4 / 8

5 The hardest example Proposition If H G and G does not divide [G : H]!, then G cannot be simple. Proof Let G act on the right cosets of H (i.e., S = G/H) by right-multiplication: φ: G Perm(S) = S n, φ(g) = the permutation that sends each Hx to Hxg. Recall that the kernel of φ is the intersection of all conjugate subgroups of H: Ker φ = x 1 Hx. x G Notice that e Ker φ H G, and Ker φ G. If Ker φ = e then φ: G S n is an embedding. But this is impossible because G does not divide S n = [G : H]!. Corollary There are no simple groups of order 4. M. Macauley (Clemson) Lecture 5.7: Finite simple groups Math 410, Modern Algebra 5 / 8

6 Theorem (classification of finite simple groups) Every finite simple group is isomorphic to one of the following groups: A cyclic group Z p, with p prime; An alternating group A n, with n 5; A Lie-type Chevalley group: PSL(n, q), PSU(n, q), PsP(n, p), and PΩ ɛ (n, q); A Lie-type group (twisted Chevalley group or the Tits group): D 4(q), E 6(q), E 7(q), E 8(q), F 4(q), F 4( n ), G (q), G (3 n ), B( n ); One of 6 exceptional sporadic groups. The two largest sporadic groups are the: baby monster group B, which has order B = ; monster group M, which has order M = The proof of this classification theorem is spread across 15,000 pages in 500 journal articles by over 100 authors, published between 1955 and 004. M. Macauley (Clemson) Lecture 5.7: Finite simple groups Math 410, Modern Algebra 6 / 8

7 q n(n+1)/ n (q i+1 1) (n+1,q 1) i=1 F but is the (index ) commutator subgroup of It is usually given honorary Lie type status. The groups starting on the second row are the classical groups. The sporadic suzuki group is unrelated to the families of Suzuki groups. Copyright c 01 Ivan Andrus q 36 (q 1 1)(q 9 1)(q 8 1) (q 6 1)(q 5 1)(q 1) (3, q 1) F q 63 9 (q i 1) (, q 1) i =, q 10 (q 30 1)(q 4 1) (q 0 1)(q 18 1)(q 14 1) (q 1 1)(q 8 1)(q 1) in the upper left are other names by which they may be known. For specific non-sporadic groups these are used to indicate isomorphims. All such isomorphisms appear on the table except the family Bn( m ) = Cn( m ). with the following exceptions: A8 = A3() and A(4) of order q 4 (q 1 1)(q 8 1) (q 6 1)(q 1) 1 q 6 (q 6 1)(q 1) q 1 (q 8 + q 4 + 1) (q 6 1)(q 1) q 36 (q 1 1)(q 9 + 1)(q 8 1) (q 6 1)(q 5 + 1)(q 1) (3, q + 1) q (q + 1)(q 1) q 1 (q 6 + 1)(q 4 1) (q 3 + 1)(q 1) q 3 (q 3 + 1)(q 1) On+1(q), Ωn+1(q) q n n (q i 1) (, q 1) i=1 q n n (q i 1) (, q 1) i=1 q n(n 1) (q n n 1 1) (q i 1) (4,q n 1) q n(n 1) (q n n 1 +1) (q i 1) (4,q n +1) q n(n+1)/ n+1 (q i ( 1) i ) (n+1,q+1) i= F7, HHM, HT H Image by Ivan Andrus, 01 The Periodic Table Of Finite Simple Groups 0, C1, Z1 1 Dynkin Diagrams of Simple Lie Algebras 1 An 1 3 n F4 C A1(4), A1(5) A5 A() A1(7) Bn 1 3 n Dn 3 4 n G A3(4) B(3) C3(3) D4() D 4 ( ) G() A (9) C3 60 A1(9), B() A6 168 G(3) A1(8) Cn 1 3 n E6,7, B(4) C3(5) D4(3) D 4 (3 ) A (16) 3 C Tits A7 A1(11) E6() E7() E8() F4() G(3) 3D 4 ( 3 ) E 6 ( ) B ( 3 ) F 4 () G (3 3 ) B3() C4(3) D5() D 5 ( ) A (5) C A3() A8 A1(13) E6(3) E7(3) E8(3) F4(3) G(4) 3D 4 (3 3 ) E 6 (3 ) B ( 5 ) F 4 ( 3 ) G (3 5 ) B(5) C3(7) D4(5) D 4 (4 ) A 3 (9) C A9 A1(17) E6(4) E7(4) E8(4) F4(4) G(5) 3D 4 (4 3 ) E 6 (4 ) B ( 7 ) F 4 ( 5 ) G (3 7 ) B(7) C3(9) D5(3) D 4 (5 ) A (64) C PSLn+1(q), Ln+1(q) PSpn(q) O + n (q) O n (q) PSUn+1(q) Zp An An(q) E6(q) E7(q) E8(q) F4(q) G(q) 3D 4 (q 3 ) E 6 (q ) B( n+1 ) F 4( n+1 ) G(3 n+1 ) Bn(q) Cn(q) Dn(q) D n(q ) A n(q ) Cp n! i=1 i=1 i=1 p Alternating Groups Classical Chevalley Groups Chevalley Groups Classical Steinberg Groups Steinberg Groups Suzuki Groups Ree Groups and Tits Group Sporadic Groups Cyclic Groups The Tits group 4 () is not a group of Lie type, 4 (). Alternates Symbol Order For sporadic groups and families, alternate names Finite simple groups are determined by their order Bn(q) and Cn(q) for q odd, n > ; M M M M M Sz O NS, O S 3 1 Suz O N Co3 Co Co1 J(1), J(11) J F5, D HN H J J LyS Ly H JM J F3, E Th J4 HS M() M(3) Fi Fi McL He F3+, M(4) F Fi 4 B Ru F1, M1 M M. Macauley (Clemson) Lecture 5.7: Finite simple groups Math 410, Modern Algebra 7 / 8

8 Finite Simple Group (of Order Two), by The Klein Four TM M. Macauley (Clemson) Lecture 5.7: Finite simple groups Math 410, Modern Algebra 8 / 8