Lecture 5.7: Finite simple groups Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 410, Modern Algebra M. Macauley (Clemson) Lecture 5.7: Finite simple groups Math 410, Modern Algebra 1 / 8
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
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 7 7 1. 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 3 3 1. 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
A harder example Proposition There are no simple groups of order 351. Proof Since G = 351 = 3 3 13, the Third Sylow Theorem tells us: n 13 divides 3 3 = 7 (so n 13 {1, 3, 9, 7}) n 13 13 1. 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 351 34 = 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
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
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 = 41 3 13 5 6 7 11 13 17 19 3 31 47 4.15 10 33 ; monster group M, which has order M = 46 3 0 5 9 7 6 11 13 3 17 19 3 9 31 41 47 59 71 8.08 10 53. 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
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. 7 57 703 347 541 463 10 08 58 395 14 643 00 85 58 710 781 34 640 103 833 619 055 14 765 466 746 880 000 q 36 (q 1 1)(q 9 1)(q 8 1) (q 6 1)(q 5 1)(q 1) (3, q 1) F 7 997 476 04 075 799 759 100 487 6 680 80 918 400 1 71 375 36 818 136 74 40 479 751 139 01 644 554 379 03 770 766 54 617 395 00 111 131 458 114 940 385 379 597 33 477 884 941 80 664 199 57 155 056 307 51 745 63 504 588 800 000 000 q 63 9 (q i 1) (, q 1) i =,8 337 804 753 143 634 806 61 388 190 614 085 595 079 991 69 4 467 651 576 160 959 909 068 800 000 18 830 05 91 953 93 311 099 03 439 97 660 33 140 886 784 940 15 038 5 449 391 86 616 580 150 109 878 711 43 949 98 163 694 448 66 40 940 800 000 191 797 9 14 671 717 754 639 757 897 51 906 41 357 507 604 16 557 533 558 87 598 36 977 154 17 870 984 484 770 345 340 348 98 409 697 395 609 8 849 49 17 656 441 474 908 160 000 000 000 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 0160. 19 009 85 53 840 945 451 97 669 10 000 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) 4 14 636 855 916 969 695 633 965 10 680 53 377 600 85 696 576 147 617 709 485 896 77 387 584 983 695 360 000 000 q 36 (q 1 1)(q 9 + 1)(q 8 1) (q 6 1)(q 5 + 1)(q 1) (3, q + 1) 1 3 4 q (q + 1)(q 1) 1 1 318 633 155 799 591 447 70 161 609 78 7 560 000 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 4 154 781 481 6 46 191 177 580 544 000 000 808 017 44 794 51 875 886 459 904 961 710 757 005 754 368 000 000 000 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 E6,7,8 1 3 5 6 7 8 5 90 B(4) 4 585 351 680 C3(5) 174 18 400 D4(3) 197 406 70 D 4 (3 ) 6 048 A (16) 3 C5 360 504 979 00 8 501 000 000 000 4 95 179 814 400 10 151 968 619 50 6 400 5 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) C7 50 660 14 841 575 5 005 575 70 400 3 311 16 603 366 400 4 45 696 11 341 31 76 53 479 683 774 853 939 00 9 10 17 971 00 10 073 444 47 1 451 50 65 784 756 654 489 600 3 499 95 948 800 5 015 379 558 400 16 000 7 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) C11 0 160 1 09 5 734 40 79 816 671 844 761 600 51 596 800 0 560 831 566 91 3 537 600 64 905 35 699 586 176 614 400 49 85 657 439 340 55 4 680 000 73 457 18 604 953 600 8 911 539 000 000 000 000 67 536 471 195 648 000 3 65 90 11 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) C13 181 440 448 5 859 000 000 67 80 350 64 790 400 34 093 383 680 39 189 910 64 35 349 33 63 138 97 600 54 05 731 40 499 584 000 1 89 51 799 941 305 139 00 17 880 03 50 000 000 000 5 515 776 13 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 > ; M11 7 90 M1 95 040 M 443 50 M3 10 00 960 M4 44 83 040 Sz O NS, O S 3 1 Suz O N Co3 Co Co1 J(1), J(11) J1 175 560 F5, D HN 73 030 91 000 000 H J J 604 800 LyS Ly 51 765 179 004 000 000 H JM J3 50 3 960 F3, E Th 90 745 943 887 87 000 J4 HS 86 775 571 046 077 56 880 44 35 000 M() M(3) Fi Fi3 4 089 470 473 64 561 751 654 400 93 004 800 McL He 898 18 000 4 030 387 00 F3+, M(4) F Fi 4 B 1 55 05 709 190 661 71 9 800 Ru 145 96 144 000 F1, M1 M M. Macauley (Clemson) Lecture 5.7: Finite simple groups Math 410, Modern Algebra 7 / 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