MATHEMATICS OF COMPUTATION VOLUME 4«, NUMBER 174 APRIL 1986, PAGES 609-611 A Conjecture of Frobenius and the Sporadic Simple Groups, II* By Hiroyoshi Yamaki Abstract. A conjecture of Frobenius which has been reduced to the classification of finite simple groups is verified for the sporadic simple groups. Let G be a finite group and « be a positive integer dividing \G\. Let Ln(G) = {x e G\x" = 1). Then by a theorem of Frobenius [6] one knows that \Ln(G)\ = cnn for some integer cn. Frobenius conjectured that Ln(G) forms a subgroup of G provided \Ln(G)\ = « (see [2]). Zemlin [25] has reduced the conjecture to the classification of finite simple groups which is now complete (see [8]). The author has verified the conjecture for the Fischer Griess monster Fx and the Fischer baby monster F2 in [24]. The purpose of this note is to prove the following Theorem. The conjecture of Frobenius is true for all the sporadic simple groups. The proof of our theorem has been carried out in the following way with the use of a computer. Let G be one of the sporadic simple groups. By [24] we may assume that G + Fx and G + F2. Let f(G, t) be the number of elements of order t in G and Ord(G) = (order of x \ x e G}. Tables of f(G, t) are given in the Appendix; see the supplements section at the end of this issue. For f(G,t) the reader is referred to the following papers: Mxx, M22, M23 Burgoyne and Fong [1] •^12' ^24 Frobenius [5] Jx Janko [14] HJ = J2 Hall and Wales [8] HJM = J3 Janko [15] JA Janko [16] HiS Frame [4] Suz Wright [23] McL, .3 Finkelstein [3] Rud Rudvalis [19] HUM Held [10] LyS Lyons [17] ReceivedJune 21, 1982; revisedMarch 19, 1984 and July 11, 1984. 1980 Mathematics Subject Classification. Primary 20D05. Key words and phrases. Simple groups, a conjecture of Frobenius. *Dedicated to Professor Nagayoshi Iwahori on his 60th birthday. ©1986 American Mathematical Society 0025-5718/86 $1.00 + $.25 per page 609 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 610 hiroyoshi yamaki ON O'Nan [18] .1 Wilson [21] .2 Wilson [22] M(22) Hunt [11] M(23) Hunt [12] M(24)' Hunt [13] F3 Smith [20] F5 Harada [9] We look for all the subsets ß of Ord(G) satisfying the following two conditions: (a) If t is a member of ß, then ß contains all divisors of t. In particular 1 is always a member of ß. (b) For the subset ß of Ord(G) in (a), E,eS2/(G, /) divides |G|. A PASCAL program effectively generates such a subset ß with the use of a recursive concept. Special add and divide routines are used for explicit calculation with very large digit numbers. Then we have only six possibilities for ß: (i) ß = {1}, (ii) ß = Ord(G), (iii) ß = (1,2,3,4,5,10,11} and G is the Mathieu group MX2, (iv) ß = (1,2,3,5,7,11,19} and G is the Janko group Jx, (v) ß = (1,2,3,4,5,6,7,8,12,13,14,26,29} and G is the Rudvalisgroup Rud, (vi) ß = {1,2,3,4,5,6,7,8,13,14,16,26,29} and G is the Rudvalisgroup Rud. It follows from the Table of f(G, t) in the Appendix that (1, ifß={l}, £/(G,í) = (|G|, ifß = Ord(G), \|G|/2, otherwise. Since « = \Ln(G)\ = Lt<=af(G, t) for some ß satisfying the two conditions (a) and (b), we have « = 1 or n = \G\. This verifies the conjecture for G. The proof of our theorem is complete. Acknowledgment. The author thanks Sachiko Yamaki for her help with the computer calculations. Department of Mathematics University of Tsukuba Ibaraki 305, Japan 1. N. Burgoyne & P. Fong, "The Schur multipliers of the Mathieu groups," Nagoya Math.J., v. 27, 1966, pp. 733-745. [Correction, ibid., v. 31,1968, pp. 297-304.] 2. W. Feit, "Some consequences of the classification of finite simple groups," Proc. Sympos. Pure Math., v. 37, 1980,pp. 175-181. 3. L. Finkelstein, "The maximal subgroups of Conway's group C3 and McLaughlin's group," J. Algebra,v. 25,1973, pp. 58-89. 4. J. S. Frame, " Computation of characters of the Higman-Sims group and its automorphism group," J. Algebra, v. 20, 1972, pp. 320-349. 5. G. Frobenius, "Über die Charaktere der mehrfach transitiven Gruppen," Berliner Ber., 1904, pp. 558-571. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A CONJECTURE OF FROBENIUS AND THE SPORADIC SIMPLE GROUPS 611 6. G Frobenius, "Über einen Fundamen talsatz der Gruppen théorie II," Berliner Ber., 1907, pp. 428-437. 7. D. GORENSTEIN,Finite Simple Groups, Plenum Press, New York, London, 1982. 8. M. Hall, Jr. & D. Wales, "The simple group of order 604,800," /. Algebra, v. 9, 1968, pp. 417-450. 9. K. Harada, "On the simple group F of order 214 • 36 • 56 ■7 • 11 ■19," Proc. Conf. on Finite Groups, Academic Press, New York, London, 1976, pp. 119-276. 10. D. Held, "The simple groups related to M24," J. Algebra, v. 13,1969, pp. 253-296. 11. D. C. Hunt, "Character tables of certain finite simple groups," Bull. Austral. Math. Soc, v. 5, 1971,pp. 1-42. 12. D. C. Hunt, "The character table of Fischer's simple group A/(23)," Math. Comp., v. 28, 1974, pp. 660-661. 13. D. C. Hunt, Computer print-out. 14. Z. Janko, "A new finite simple group with abelian Sylow 2-subgroups and its characterization," J. Algebra,v. 3,1966, pp. 147-186. 15. Z. Janko, "Some new simple groups of finite order I," Symposia Math. (INDAM, Rome, 1967/68), Vol. 1, Academic Press, London, 1969, pp. 25-64. 16. Z. Janko, "A new finite simple group of order 86 • 775 ■571 ■046 • 007 ■562 • 880 which possesses M2¿ and the full covering group of M22 as subgroups," J. Algebra, v. 42,1976, pp. 564-596. 17. R. Lyons, "Evidence for a new finite simple group," J. Algebra, v. 20,1972, pp. 540-569. 18. M. E. O'Nan, "Some evidence for the existence of a new simple group," Proc. London Math. Soc, v. 32,1976, pp. 421-479. 19. A. Rudvalis, "A rank 3 simple group of order 214■ 33 ■53 • 7 ■13 ■29 I, II," J. Algebra, v. 86, 1984,pp. 181-218,219-258. 20. P. E. Smith, "A simple subgroup of Ml and £8(3)," Bull. London Math. Soc, v. 8, 1976, pp. 161-165. 21. R. A. Wilson, "The maximal subgroups of Conway's group Co,," J. Algebra, v. 85, 1983, pp. 144-165. 22. R. A. Wilson, "The maximal subgroups of Conway's group .2," J. Algebra, v. 84, 1983, pp. 107-114. 23. D. Wright, "The irreducible characters of the simple group of M. Suzuki of order 448, 345, 497, 600," J. Algebra,v. 29,1974, pp. 303-323. 24. H. Yamaki, "A conjecture of Frobenius and the sporadic simple groups," Comm. Algebra, v. 11, 1983,pp. 2513-2518. 25. R. Zemlin, On a Conjecture Arising from a Theorem of Frobenius, Ph. D. Thesis, Ohio State Univeristy, 1954. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use MATHEMATICS OF COMPUTATION VOLUME 46. NUMBER 174 APRIL 1986, PAGES S43-S46 Supplement to A Conjecture of Frobenius and the Sporadic Simple Groups. II By Hiroyoshi Yamaki Appendix The following is the Table of f(0,t) for the sporadic simple groups we study. "-" means f(O,t)=0, t »il V12 K22 K23 K2h Jl 1111 1 11 S 165 891 1,155 3,795 U3.263 1.U63 3 Uo l4,li00 12,320 56,672 712,1*1)6 5,852 ■i 990 5,9>tO 141,580 318,780 5,100,1480 5 l,581i 9,50l4 88,7014 68o,o6l4 14,080,38I4 ll,70li 6 1,320 23,760 36,960 850,080 20,1401,920 29,260 7 - - 126,720 1,1457,280 11,658,2140 25,080 8 1,980 23,760 55,>4l40 1,275,120 15,301,M40 10 - 9,50l4 - - 12,2l4l,152 35,112 11 l.UltO 17,280 80,61*0 1,8514,720 22,256,6140 15,960 12 - 1,0,803,8140 114 - 1,1457,280 314,9714,720 15 - - - 1,360,128 32,6143,072 23,1408 19 - - 27,720 21 - 23,316,1480 23 - - - 887,0140 21,288,960 t HJ HJK HiS McL 1111 1 2 2,835 26,163 21,175 22,275 3 17,360 253,232 123,200 9514,800 I4 6,300 523,260 877,800 9,355,500 5 28,22>4 3,3l48,86l4 2,010,6214 37,122,6214 6 75,600 2,093,0140 3,080,000 27,!4¡42,800 7 86,1,00 - 6,336,000 128,3014,000 8 75,600 6,279,120 8,316,000 112,266,000 9 - 5,581,1*1*0 - 66,528,000 10 181,1*1*0 10,01*6,592 1*,1*35,200 29,937,600 11 8,o6i* ,000 163,296,000 12 50,1*00 1*,186,080 3,696,000 7I*,81*1* ,000 11* - - 128,301* ,000 15 80,61*0 6,697,728 2,956,800 59,875,200 17 - 5,909,760 19 - 5,287,680 20 - - 1*,1*35,200 30 - - - 59,875,200 ©1986 American Mathematical Society 0025-5718/86 $1.00 + $.25 per page S43 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use S44 SUPPLEMENT Ov CM O VO _ O -X O VO CO VO O VO 00 CO vü On VO CM VO CO r-H 00 -3 CO VO CO O co -3- vu CM Ov vo cO co .3 la co co on ov Ov CM Ov w u-\o m j vo en la co vo ovo cvi^ co ov t— -3 IA H CM co la ov vo ío co Ov vo CO O O CO (H eo H md m o on cu ia o t— o o\ CM Ov CO fli pj iA (ti Ctj IA (\j VD t- lA O H -3 ON-3- m ia vo ia vo o co ia o la on Ov lA MO ovocoocoocmo on ov la o cm vo vo Q\ i-i co la 0j cmcocmocmcm-3co OCO O H CM CM Ov O V£> la t— CM On CO CO co w U) co co .3 ro cm VO CO CO vO CO (\) J CO O O la ro m m ov LA .3 O Oí 1*1 iA 0\ t- CO 4 la co cm o\ 0\ 0\ H m m co Rl(M(\lCJ>iAtCO\W&W Ov lA Ov On Ov -3^ Ov t"- W O 4 CV 00 On on O O CO ■a co lA on co o\ ov CO CO VO VO CJ\ IA W 5 ÍÜCM H CO CO t— CO CO VO CO CM CO CM O CO C\j \0 Ov CO CO cu m t- j en CO VO VO CO _3 CM OV C— t— 0J M3 CO CM CM VO CM CO O VO CM CM LA VO VO CO VO On vO 0\ vx» m On ia On On CO iH MD la LA VO OV VO co o\ cm co vo ON, CO 1A t— c— o\ on la on IA VO t- CO ON O _ h ni m j vo œ o IA V£> OV O CMCMCMCMCMCMCMCO ro on ro „3 On CM CM LA VO LA ON ON LACOO VO On VO t-cOHvovo t— h on ON CO t— t— o c\i cm ov t— en .3 VO VO CO CM CM lA O O O O -3 O VO O -3- O H CO la -a- r- CO O CO CO CM VU Ov CO n h w o co m VO CM J- CM IA VO CM LACOVOcOcO CM H CM CO H CO t- t- MO Ov CM on LA co on ov vo CM H -3 CO VO i—1 LA LA CO ON VO CMVO r- C\ r-i la o -3 co m lA ri