DOCSLIB.ORG

Finite Simple Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Finite Simple Groups 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.
Load more

Recommended publications
  • Math 412. Simple Groups
    Math 412. Simple Groups DEFINITION: A group G is simple if its only normal subgroups are feg and G. Simple groups are rare among all groups in the same way that prime numbers are rare among all integers. The smallest non-abelian group is A5, which has order 60. THEOREM 8.25: A abelian group is simple if and only if it is finite of prime order. THEOREM: The Alternating Groups An where n ≥ 5 are simple. The simple groups are the building blocks of all groups, in a sense similar to how all integers are built from the prime numbers. One of the greatest mathematical achievements of the Twentieth Century was a classification of all the finite simple groups. These are recorded in the Atlas of Simple Groups. The mathematician who discovered the last-to-be-discovered finite simple group is right here in our own department: Professor Bob Greiss. This simple group is called the monster group because its order is so big—approximately 8 × 1053. Because we have classified all the finite simple groups, and we know how to put them together to form arbitrary groups, we essentially understand the structure of every finite group. It is difficult, in general, to tell whether a given group G is simple or not. Just like determining whether a given (large) integer is prime, there is an algorithm to check but it may take an unreasonable amount of time to run. A. WARM UP. Find proper non-trivial normal subgroups of the following groups: Z, Z35, GL5(Q), S17, D100.
    [Show full text]
  • Chapter 25 Finite Simple Groups
    Chapter 25 Finite Simple Groups Chapter 25 Finite Simple Groups Historical Background Definition A group is simple if it has no nontrivial proper normal subgroup. The definition was proposed by Galois; he showed that An is simple for n ≥ 5 in 1831. It is an important step in showing that one cannot express the solutions of a quintic equation in radicals. If possible, one would factor a group G as G0 = G, find a normal subgroup G1 of maximum order to form G0/G1. Then find a maximal normal subgroup G2 of G1 and get G1/G2, and so on until we get the composition factors: G0/G1,G1/G2,...,Gn−1/Gn, with Gn = {e}. Jordan and Hölder proved that these factors are independent of the choices of the normal subgroups in the process. Jordan in 1870 found four infinite series including: Zp for a prime p, SL(n, Zp)/Z(SL(n, Zp)) except when (n, p) = (2, 2) or (2, 3). Between 1982-1905, Dickson found more infinite series; Miller and Cole showed that 5 (sporadic) groups constructed by Mathieu in 1861 are simple. Chapter 25 Finite Simple Groups In 1950s, more infinite families were found, and the classification project began. Brauer observed that the centralizer has an order 2 element is important; Feit-Thompson in 1960 confirmed the 1900 conjecture that non-Abelian simple group must have even order. From 1966-75, 19 new sporadic groups were found. Thompson developed many techniques in the N-group paper. Gorenstein presented an outline for the classification project in a lecture series at University of Chicago in 1972.
    [Show full text]
  • Prescribed Gauss Decompositions for Kac-Moody Groups Over Fields
    RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA JUN MORITA EUGENE PLOTKIN Prescribed Gauss decompositions for Kac- Moody groups over fields Rendiconti del Seminario Matematico della Università di Padova, tome 106 (2001), p. 153-163 <http://www.numdam.org/item?id=RSMUP_2001__106__153_0> © Rendiconti del Seminario Matematico della Università di Padova, 2001, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Prescribed Gauss Decompositions for Kac-Moody Groups Over Fields. JUN MORITA(*) - EUGENE PLOTKIN (**) ABSTRACT - We obtain the Gauss decomposition with prescribed torus elements for a Kac-Moody group over a field containing sufficiently many elements. 1. Introduction. Kac-Moody groups are equipped with canonical decompositions of different types. Let us note, for instance, the decompositions of Bruhat, Birkhoff and Gauss. As in the finite dimensional case, they play an im- portant role in calculations with these groups. However, in the Kac-Moo- dy case each of these decompositions has its own special features (see, for example, [18] where J. Tits compares the Bruhat and the Birkhoff decompositions and presents some applications). The aim of this paper is to establish the so-called prescribed Gauss decomposition for Kac-Moody groups.
    [Show full text]
  • A NEW APPROACH to UNRAMIFIED DESCENT in BRUHAT-TITS THEORY by Gopal Prasad
    A NEW APPROACH TO UNRAMIFIED DESCENT IN BRUHAT-TITS THEORY By Gopal Prasad Dedicated to my wife Indu Prasad with gratitude Abstract. We present a new approach to unramified descent (\descente ´etale") in Bruhat-Tits theory of reductive groups over a discretely valued field k with Henselian valuation ring which appears to be conceptually simpler, and more geometric, than the original approach of Bruhat and Tits. We are able to derive the main results of the theory over k from the theory over the maximal unramified extension K of k. Even in the most interesting case for number theory and representation theory, where k is a locally compact nonarchimedean field, the geometric approach described in this paper appears to be considerably simpler than the original approach. Introduction. Let k be a field given with a nontrivial R-valued nonarchimedean valuation !. We will assume throughout this paper that the valuation ring × o := fx 2 k j !(x) > 0g [ f0g of this valuation is Henselian. Let K be the maximal unramified extension of k (the term “unramified extension" includes the condition that the residue field extension is separable, so the residue field of K is the separable closure κs of the residue field κ of k). While discussing Bruhat-Tits theory in this introduction, and beginning with 1.6 everywhere in the paper, we will assume that ! is a discrete valuation. Bruhat- Tits theory of connected reductive k-groups G that are quasi-split over K (i.e., GK contains a Borel subgroup, or, equivalently, the centralizer of a maximal K-split torus of GK is a torus) has two parts.
    [Show full text]
  • CURTIS-TITS GROUPS GENERALIZING KAC-MOODY GROUPS of TYPE Aen−1
    CURTIS-TITS GROUPS GENERALIZING KAC-MOODY GROUPS OF TYPE Aen−1 RIEUWERT J. BLOK AND CORNELIU G. HOFFMAN Abstract. In [13] we define a Curtis-Tits group as a certain generalization of a Kac- Moody group. We distinguish between orientable and non-orientable Curtis-Tits groups and identify all orientable Curtis-Tits groups as Kac-Moody groups associated to twin- buildings. In the present paper we construct all orientable as well as non-orientable Curtis-Tits groups with diagram Aen−1 (n ≥ 4) over a field k of size at least 4. The resulting groups are quite interesting in their own right. The orientable ones are related to Drinfeld’s construction of vector bundles over a non-commutative projective line and to the classical groups over cyclic algebras. The non-orientable ones are related to expander graphs [14] and have symplectic, orthogonal and unitary groups as quotients. Keywords: Curtis-Tits groups, Kac-Moody groups, Moufang, twin-building, amalgam, opposite. MSC 2010: 20G35 51E24 1. Introduction The theory of the infinite dimensional Lie algebras called Kac-Moody algebras was ini- tially developed by Victor Kac and Robert Moody. The development of a theory of Kac- Moody groups as analogues of Chevalley groups was made possible by the work of Kac and Peterson. In [44] J. Tits gives an alternative definition of a group of Kac-Moody type as being a group with a twin-root datum, which implies that they are symmetry groups of Moufang twin-buildings. In [2] P. Abramenko and B. Mühlherr generalize a celebrated theorem of Curtis and Tits on groups with finite BN-pair [18, 42] to groups of Kac-Moody type.
    [Show full text]
  • Quasi P Or Not Quasi P? That Is the Question
    Rose-Hulman Undergraduate Mathematics Journal Volume 3 Issue 2 Article 2 Quasi p or not Quasi p? That is the Question Ben Harwood Northern Kentucky University, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Harwood, Ben (2002) "Quasi p or not Quasi p? That is the Question," Rose-Hulman Undergraduate Mathematics Journal: Vol. 3 : Iss. 2 , Article 2. Available at: https://scholar.rose-hulman.edu/rhumj/vol3/iss2/2 Quasi p- or not quasi p-? That is the Question.* By Ben Harwood Department of Mathematics and Computer Science Northern Kentucky University Highland Heights, KY 41099 e-mail: [email protected] Section Zero: Introduction The question might not be as profound as Shakespeare’s, but nevertheless, it is interesting. Because few people seem to be aware of quasi p-groups, we will begin with a bit of history and a definition; and then we will determine for each group of order less than 24 (and a few others) whether the group is a quasi p-group for some prime p or not. This paper is a prequel to [Hwd]. In [Hwd] we prove that (Z3 £Z3)oZ2 and Z5 o Z4 are quasi 2-groups. Those proofs now form a portion of Proposition (12.1) It should also be noted that [Hwd] may also be found in this journal. Section One: Why should we be interested in quasi p-groups? In a 1957 paper titled Coverings of algebraic curves [Abh2], Abhyankar conjectured that the algebraic fundamental group of the affine line over an algebraically closed field k of prime characteristic p is the set of quasi p-groups, where by the algebraic fundamental group of the affine line he meant the family of all Galois groups Gal(L=k(X)) as L varies over all finite normal extensions of k(X) the function field of the affine line such that no point of the line is ramified in L, and where by a quasi p-group he meant a finite group that is generated by all of its p-Sylow subgroups.
    [Show full text]
  • FINITE GROUP ACTIONS on REDUCTIVE GROUPS and BUILDINGS and TAMELY-RAMIFIED DESCENT in BRUHAT-TITS THEORY by Gopal Prasad Dedicat
    FINITE GROUP ACTIONS ON REDUCTIVE GROUPS AND BUILDINGS AND TAMELY-RAMIFIED DESCENT IN BRUHAT-TITS THEORY By Gopal Prasad Dedicated to Guy Rousseau Abstract. Let K be a discretely valued field with Henselian valuation ring and separably closed (but not necessarily perfect) residue field of characteristic p, H a connected reductive K-group, and Θ a finite group of automorphisms of H. We assume that p does not divide the order of Θ and Bruhat-Tits theory is available for H over K with B(H=K) the Bruhat-Tits building of H(K). We will show that then Bruhat-Tits theory is also available for G := (HΘ)◦ and B(H=K)Θ is the Bruhat-Tits building of G(K). (In case the residue field of K is perfect, this result was proved in [PY1] by a different method.) As a consequence of this result, we obtain that if Bruhat-Tits theory is available for a connected reductive K-group G over a finite tamely-ramified extension L of K, then it is also available for G over K and B(G=K) = B(G=L)Gal(L=K). Using this, we prove that if G is quasi-split over L, then it is already quasi-split over K. Introduction. This paper is a sequel to our recent paper [P2]. We will assume fa- miliarity with that paper; we will freely use results, notions and notations introduced in it. Let O be a discretely valued Henselian local ring with valuation !. Let m be the maximal ideal of O and K the field of fractions of O.
    [Show full text]
  • Title: Algebraic Group Representations, and Related Topics a Lecture by Len Scott, Mcconnell/Bernard Professor of Mathemtics, the University of Virginia
    Title: Algebraic group representations, and related topics a lecture by Len Scott, McConnell/Bernard Professor of Mathemtics, The University of Virginia. Abstract: This lecture will survey the theory of algebraic group representations in positive characteristic, with some attention to its historical development, and its relationship to the theory of finite group representations. Other topics of a Lie-theoretic nature will also be discussed in this context, including at least brief mention of characteristic 0 infinite dimensional Lie algebra representations in both the classical and affine cases, quantum groups, perverse sheaves, and rings of differential operators. Much of the focus will be on irreducible representations, but some attention will be given to other classes of indecomposable representations, and there will be some discussion of homological issues, as time permits. CHAPTER VI Linear Algebraic Groups in the 20th Century The interest in linear algebraic groups was revived in the 1940s by C. Chevalley and E. Kolchin. The most salient features of their contributions are outlined in Chapter VII and VIII. Even though they are put there to suit the broader context, I shall as a rule refer to those chapters, rather than repeat their contents. Some of it will be recalled, however, mainly to round out a narrative which will also take into account, more than there, the work of other authors. §1. Linear algebraic groups in characteristic zero. Replicas 1.1. As we saw in Chapter V, §4, Ludwig Maurer thoroughly analyzed the properties of the Lie algebra of a complex linear algebraic group. This was Cheval­ ey's starting point.
    [Show full text]
  • The Mathieu Groups (Simple Sporadic Symmetries)
    The Mathieu Groups (Simple Sporadic Symmetries) Scott Harper (University of St Andrews) Tomorrow's Mathematicians Today 21st February 2015 Scott Harper The Mathieu Groups 21st February 2015 1 / 15 The Mathieu Groups (Simple Sporadic Symmetries) Scott Harper (University of St Andrews) Tomorrow's Mathematicians Today 21st February 2015 Scott Harper The Mathieu Groups 21st February 2015 2 / 15 1 2 A symmetry is a structure preserving permutation of the underlying set. A group acts faithfully on an object if it is isomorphic to a subgroup of the 4 3 symmetry group of the object. Symmetry group: D4 The stabiliser of a point in a group G is Group of rotations: the subgroup of G which fixes x. ∼ h(1 2 3 4)i = C4 Subgroup fixing 1: h(2 4)i Symmetry Scott Harper The Mathieu Groups 21st February 2015 3 / 15 A symmetry is a structure preserving permutation of the underlying set. A group acts faithfully on an object if it is isomorphic to a subgroup of the symmetry group of the object. The stabiliser of a point in a group G is Group of rotations: the subgroup of G which fixes x. ∼ h(1 2 3 4)i = C4 Subgroup fixing 1: h(2 4)i Symmetry 1 2 4 3 Symmetry group: D4 Scott Harper The Mathieu Groups 21st February 2015 3 / 15 A group acts faithfully on an object if it is isomorphic to a subgroup of the symmetry group of the object. The stabiliser of a point in a group G is Group of rotations: the subgroup of G which fixes x.
    [Show full text]
  • 2,3,T)-Generations for the Tits Simple Group 2F4(2
    2 0 (2; 3; t)-Generations for the Tits simple group F4(2) MOHAMMED ALI FAYA IBRAHIM FARYAD ALI King Khalid University King Khalid University Department of Mathematics Department of Mathematics P.O. Box 9004, Abha P.O. Box 9004, Abha SAUDI ARABIA SAUDI ARABIA mafi[email protected] [email protected] Abstract: A group G is said to be (2; 3; t)-generated if it can be generated by two elements x and y such that o(x) = 2, o(y) = 3 and o(xy) = t. In this paper, we determine (2; 3; t)-generations of the Tits simple group ∼ 2 0 T = F4(2) where t is divisor of jTj. Most of the computations were carried out with the aid of computer algebra system GAP [17]. 2 0 Key–Words: Tits group F4(2) , simple group, (2; 3; t)-generation, generator. 1 Introduction was first determined by Tchakerian [19]. Later but independently, Wilson [20] also determined the max- A group G is called (2; 3; t)-generated if it can be gen- imal subgroups of the simple group T, while studying erated by an involution x and an element y of order 3 the geometry of the simple groups of Tits and Rud- such that o(xy) = t. The (2; 3)-generation problem valis. has attracted a vide attention of group theorists. One For basic properties of the Tits group T and infor- reason is that (2; 3)-generated groups are homomor- mation on its subgroups the reader is referred to [20], phic images of the modular group P SL(2; Z), which [19].
    [Show full text]
  • A FRIENDLY INTRODUCTION to GROUP THEORY 1. Who Cares?
    A FRIENDLY INTRODUCTION TO GROUP THEORY JAKE WELLENS 1. who cares? You do, prefrosh. If you're a math major, then you probably want to pass Math 5. If you're a chemistry major, then you probably want to take that one chem class I heard involves representation theory. If you're a physics major, then at some point you might want to know what the Standard Model is. And I'll bet at least a few of you CS majors care at least a little bit about cryptography. Anyway, Wikipedia thinks it's useful to know some basic group theory, and I think I agree. It's also fun and I promise it isn't very difficult. 2. what is a group? I'm about to tell you what a group is, so brace yourself for disappointment. It's bound to be a somewhat anticlimactic experience for both of us: I type out a bunch of unimpressive-looking properties, and a bunch of you sit there looking unimpressed. I hope I can convince you, however, that it is the simplicity and ordinariness of this definition that makes group theory so deep and fundamentally interesting. Definition 1: A group (G; ∗) is a set G together with a binary operation ∗ : G×G ! G satisfying the following three conditions: 1. Associativity - that is, for any x; y; z 2 G, we have (x ∗ y) ∗ z = x ∗ (y ∗ z). 2. There is an identity element e 2 G such that 8g 2 G, we have e ∗ g = g ∗ e = g. 3. Each element has an inverse - that is, for each g 2 G, there is some h 2 G such that g ∗ h = h ∗ g = e.
    [Show full text]
  • 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.
    [Show full text]