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Lecture 5.7: Finite simple groups Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 5.7: Finite simple groups Math 4120, Modern Algebra 1 / 8 Overview Definition A group G is simple if its only normal subgroups are G and hei. 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 np = 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 np = 1 for some p dividing jGj. Since we established n5 = 1 for our running example of a group of size jMj = 200 = 23 · 52, there are no simple groups of order 200. M. Macauley (Clemson) Lecture 5.7: Finite simple groups Math 4120, Modern Algebra 2 / 8 An easy example Tip When trying to show that np = 1, it's usually more helpful to analyze the largest primes first. Proposition There are no simple groups of order 84. Proof Since jGj = 84 = 22 · 3 · 7, the Third Sylow Theorem tells us: 2 n7 divides 2 · 3 = 12 (so n7 2 f1; 2; 3; 4; 6; 12g) n7 ≡7 1. The only possibility is that n7 = 1, so the Sylow 7-subgroup must be normal. Observe why it is beneficial to use the largest prime first: 2 n3 divides 2 · 7 = 28 and n3 ≡3 1. Thus n3 2 f1; 2; 4; 7; 14; 28g. n2 divides 3 · 7 = 21 and n2 ≡2 1. Thus n2 2 f1; 3; 7; 21g. M. Macauley (Clemson) Lecture 5.7: Finite simple groups Math 4120, Modern Algebra 3 / 8 A harder example Proposition There are no simple groups of order 351. Proof Since jGj = 351 = 33 · 13, the Third Sylow Theorem tells us: 3 n13 divides 3 = 27 (so n13 2 f1; 3; 9; 27g) n13 ≡13 1. The only possibilies are n13 = 1 or 27. A Sylow 13-subgroup P has order 13, and a Sylow 3-subgroup Q has order 33 = 27. Therefore, P \ Q = feg. Suppose n13 = 27. Every Sylow 13-subgroup contains 12 non-identity elements, and so G must contain 27 · 12 = 324 elements of order 13. This leaves 351 − 324 = 27 elements in G not of order 13. Thus, G contains only one Sylow 3-subgroup (i.e., n3 = 1) and so G cannot be simple. M. Macauley (Clemson) Lecture 5.7: Finite simple groups Math 4120, Modern Algebra 4 / 8 The hardest example Proposition If H G and jGj 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) =∼ Sn ; 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: \ −1 Ker φ = x Hx: x2G Notice that hei ≤ Ker φ ≤ H G, and Ker φ C G. If Ker φ = hei then φ: G ,! Sn is an embedding. But this is impossible because jGj does not divide jSnj = [G : H]!. Corollary There are no simple groups of order 24. M. Macauley (Clemson) Lecture 5.7: Finite simple groups Math 4120, 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 Zp, with p prime; An alternating group An, with n ≥ 5; A Lie-type Chevalley group: PSL(n; q), PSU(n; q), PsP(2n; p), and PΩ(n; q); A Lie-type group (twisted Chevalley group or the Tits group): D4(q), E6(q), 2 n 0 2 n 2 n E7(q), E8(q), F4(q), F4(2 ) , G2(q), G2(3 ), B(2 ); One of 26 exceptional \sporadic groups." The two largest sporadic groups are the: \baby monster group" B, which has order jBj = 241 · 313 · 56 · 72 · 11 · 13 · 17 · 19 · 23 · 31 · 47 ≈ 4:15 × 1033; \monster group" M, which has order jMj = 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 ≈ 8:08 × 1053: The proof of this classification theorem is spread across ≈ 15;000 pages in ≈ 500 journal articles by over 100 authors, published between 1955 and 2004. M. Macauley (Clemson) Lecture 5.7: Finite simple groups Math 4120, Modern Algebra 6 / 8 Image by Ivan Andrus, 2012 The Periodic Table Of Finite Simple Groups 0, C1, Z1 1 Dynkin Diagrams of Simple Lie Algebras C2 1 An F4 1 2 3 n 1 1 2 i 3 4 2 Dn 2 A1(4), A1(5) A2(2) 3 4 n A3(4) G2(2)0 2 2 2 A A (7) Bn G2 B (3) C (3) D (2) D (2 ) A (9) C 5 1 1 h 2 3 n 2 1 i 2 2 3 4 4 2 3 4 60 168 25 920 4 585 351 680 174 182 400 197 406 720 6 048 3 2 Cn E6,7,8 A1(9), B2(2)0 G2(3)0 1 i 2 3 n 1 2 3 5 6 7 8 2 2 2 A6 A1(8) B2(4) C3(5) D4(3) D4(3 ) A2(16) C5 228 501 360 504 979 200 000 000 000 4 952 179 814 400 10 151 968 619 520 62 400 5 Tits∗ 3 3 2 2 2 3 2 2 3 2 2 2 A7 A1(11) E6(2) E7(2) E8(2) F4(2) G2(3) D4(2 ) E6(2 ) B2(2 ) F4(2)0 G2(3 ) B3(2) C4(3) D5(2) D5(2 ) A2(25) C7 7 997 476 042 214 841 575 522 337 804 753 143 634 806 261 3 311 126 76 532 479 683 65 784 756 075 799 759 100 487 388 190 614 085 595 079 991 692 242 2 520 660 005 575 270 400 262 680 802 918 400 467 651 576 160 959 909 068 800 000 603 366 400 4 245 696 211 341 312 774 853 939 200 29 120 17 971 200 10 073 444 472 1 451 520 654 489 600 23 499 295 948 800 25 015 379 558 400 126 000 7 A3(2) 3 3 2 2 2 5 2 3 2 5 2 2 2 A8 A1(13) E6(3) E7(3) E8(3) F4(3) G2(4) D4(3 ) E6(3 ) B2(2 ) F4(2 ) G2(3 ) B2(5) C3(7) D4(5) D4(4 ) A3(9) C11 1 271 375 236 818 136 742 240 18 830 052 912 953 932 311 099 032 439 5 734 420 792 816 264 905 352 699 49 825 657 273 457 218 8 911 539 000 67 536 471 7 257 703 347 541 463 210 479 751 139 021 644 554 379 972 660 332 140 886 784 940 152 038 522 14 636 855 916 969 695 633 449 391 826 616 580 150 109 878 711 243 20 160 1 092 028 258 395 214 643 200 203 770 766 254 617 395 200 949 982 163 694 448 626 420 940 800 000 671 844 761 600 251 596 800 20 560 831 566 912 965 120 680 532 377 600 32 537 600 586 176 614 400 439 340 552 4 680 000 604 953 600 000 000 000 195 648 000 3 265 920 11 3 3 2 2 2 7 2 5 2 7 2 2 2 A9 A1(17) E6(4) E7(4) E8(4) F4(4) G2(5) D4(4 ) E6(4 ) B2(2 ) F4(2 ) G2(3 ) B2(7) C3(9) D5(3) D4(5 ) A2(64) C13 85 528 710 781 342 640 191 797 292 142 671 717 754 639 757 897 85 696 576 147 617 709 1 318 633 155 111 131 458 114 940 385 379 597 233 512 906 421 357 507 604 216 557 533 558 67 802 350 239 189 910 264 54 025 731 402 1 289 512 799 17 880 203 250 103 833 619 055 142 287 598 236 977 154 127 870 984 484 770 19 009 825 523 840 945 485 896 772 387 584 799 591 447 702 161 477 884 941 280 664 199 527 155 056 345 340 348 298 409 697 395 609 822 849 181 440 2 448 765 466 746 880 000 307 251 745 263 504 588 800 000 000 492 217 656 441 474 908 160 000 000 000 451 297 669 120 000 5 859 000 000 642 790 400 983 695 360 000 000 34 093 383 680 609 782 722 560 000 352 349 332 632 138 297 600 499 584 000 941 305 139 200 000 000 000 5 515 776 13 PSLn+1(q), Ln+1(q) O2n+1(q), Ω2n+1(q) + Z PSp2n(q) O2n(q) O2−n(q) PSUn+1(q) p 3 3 2 2 2 ( 2n+1) 2 ( 2n+1) 2 2n+1 2 2 2 2 C An An(q) E6(q) E7(q) E8(q) F4(q) G2(q) D4(q ) E6(q ) B2 2 F4 2 G2(3 ) Bn(q) Cn(q) Dn(q) Dn(q ) An(q ) p 36 12 9 8 63 9 120 30 24 36 12 9 8 2 2 q (q 1)(q 1)(q 1) q 2i q (q 1)(q 1) q (q 1)(q + 1)(q 1) n n n n ( ) n 1 n(n+1)/2 n 12 6 4 qn n 1 (qn 1) n(n 1) n n 1 n(n+1)/2 n+1 n! q + 6 − 5 − 2 − ∏ (q 1) 20 −18 −14 24 12 8 12 8 4 6 − 5 2 − q 2i q 2i − − 2i q − (q +1) − q (qi 1 1) (q 1)(q 1)(q 1) (2, q 1) − (q 1)(q 1)(q 1) q (q 1)(q 1) q (q + q + 1) (q 1)(q + 1)(q 1) q (q + 1)(q 1) (q 1) (q 1) − (q 1) (q2i 1) (qi ( 1)i) ∏ − − − − i=1 12− 8 − 2 − 6 − 2 − 6 6 2 6 2 − − 2 2 3 − 3 3 ∏ ∏ ( n ) ∏ n ∏ ∏ (n+1,q 1) = − (3, q 1) i=2,8 (q 1)(q 1)(q 1) (q 1)(q 1) q (q 1)(q 1) (q 1)(q 1) (3, q + 1) q (q + 1)(q 1) (q + 1)(q 1) q (q + 1)(q 1) (2, q 1) = − (2, q 1) = − 4,q 1 = − (4,q +1) = − (n+1,q+1) = − − p 2 − i 1 − 6 − − − − − − − − − − − − − i 1 − i 1 − i 1 i 1 i 2 Alternating Groups Classical Chevalley Groups Alternates† J(1), J(11) HJ HJM F7, HHM, HTH Chevalley Groups Classical Steinberg Groups Symbol M11 M12 M22 M23 M24 J1 J2 J3 J4 HS McL He Ru Steinberg Groups 86 775 571 046 Suzuki Groups Order‡ 7 920 95 040 443 520 10 200 960 244 823 040 175 560 604 800 50 232 960 077 562 880 44 352 000 898 128 000 4 030 387 200 145 926 144 000 Ree Groups and Tits Group∗ Sporadic Groups Cyclic Groups †For sporadic groups and families, alternate names in the upper left are other names by which they may be known.
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  • Chapter 1 GENERAL STRUCTURE and PROPERTIES

    Chapter 1 GENERAL STRUCTURE and PROPERTIES

    Chapter 1 GENERAL STRUCTURE AND PROPERTIES 1.1 Introduction In this Chapter we would like to introduce the main de¯nitions and describe the main properties of groups, providing examples to illustrate them. The detailed discussion of representations is however demanded to later Chapters, and so is the treatment of Lie groups based on their relation with Lie algebras. We would also like to introduce several explicit groups, or classes of groups, which are often encountered in Physics (and not only). On the one hand, these \applications" should motivate the more abstract study of the general properties of groups; on the other hand, the knowledge of the more important and common explicit instances of groups is essential for developing an e®ective understanding of the subject beyond the purely formal level. 1.2 Some basic de¯nitions In this Section we give some essential de¯nitions, illustrating them with simple examples. 1.2.1 De¯nition of a group A group G is a set equipped with a binary operation , the group product, such that1 ¢ (i) the group product is associative, namely a; b; c G ; a (b c) = (a b) c ; (1.2.1) 8 2 ¢ ¢ ¢ ¢ (ii) there is in G an identity element e: e G such that a e = e a = a a G ; (1.2.2) 9 2 ¢ ¢ 8 2 (iii) each element a admits an inverse, which is usually denoted as a¡1: a G a¡1 G such that a a¡1 = a¡1 a = e : (1.2.3) 8 2 9 2 ¢ ¢ 1 Notice that the axioms (ii) and (iii) above are in fact redundant.