COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

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Character Table 2 3 4 5 6 7 8 1 Class 1463 5852 5852 5852 9240 9240 9240 Order 2 3 5 5 19 19 19 Period 1 1 1 1: 1 1 1 -1 -1 0 0 1 1 1 -1 -1 2+4cl 2+4c2 2: 56 0 2 -1 0 0 -1 0 0 2+4c2 3: 56 0 2 1 4: 76 4 5: 76 -4 1 2 6: 77 5 -1 1 7: 77 -3 2 1 1 8: 77 -3 2 2c1+2c7+2c8 2c4 + 2c6 + 2c9 2c2 + 2c3 + 2c5 9: 120 0 0 2c4+2c6+2c9 2c2+2c3+2c5 2c1+2c7+2c8 2c2 2c1 0 0 0 -2 2+4c1 1 1 2 2cl 2c2 0 0 0 -2 10: 120 0 0 2c2 + 2c3 + 2c5 0 0 0 0 2c1+2c7+2c8 2c4+2c6+2c9 11: 120 0 0 0 0 0 0 0 0 0 12: 133 5 1 1+2c1 1+2c2 13: 133 -3 -2 1+2c2 -1 1+2c1 -1 14: 133 -3 -2 0 15: 209 1 -1 Character Table Class 9 10 11 12 13 14 15 Order 11704 11704 15960 17556 17556 25080 29260 Period 15 15 11 10 10 7 6 1: 1 1 1 1 1 1 2: 2c3 2c6 1 0 0 0 3: 2c6 2c3 1 0 0 0 4: 1 -1 -1 -1 -1 5: 1 1 -1 1 1 -1 -1 6: -1 -1 0 0 0 0 -1 7: 2c3 2c6 0 2c2 2c4 0 0 8: 2c6 2c3 0 2c4 2c2 0 0 9: 0 0 -1 0 0 1 0 10: 0 0 -1 0 0 0 11: 0 0 -1 0 0 1 0 12: 1 1 0 0 0 -1 13: 1+2c6 1+2c3 1 2c2 2c4 0 0 14: 1+2c3 1+2c6 1 2c4 2c2 0 0 15: -1 -1 0 1 1 -1 1
100 John McKay In the table, c indicates the cosine of a multiple of 2nlperiod. For example, 2+4c2, occurring in the second row of the table as the character of an element of period 5 in the fourth conjugacy class, is an abbreviation for 2+4 cos (2X&/5). REFERENCES i. J. S. FRAME: The constructive reduction of finite group representations. Proc. Symp. Pure Maths. (AMS) 6 (1962), 89-99. 2. N. S. MENDELSOHN: Defining relations for subgroups of finite index of groups with a tinite presentation. These Proceedings, pp. 43-44. 3. J. G. F. FRANCIS: The QR transformation. Pts. 1 & 2. Computer Journal 4 (1961- 1962), 265-271, 332-345. 4. H. S. M. C~XETER~~~ W. 0. J. MOSER: Generators and Relations for Discrete Groups. Ergebnisse der Mathematik NF 14 (Springer, Berlin 1965). 5. M. HALL and J. K. SENIOR: Groups of Order 2” (n&) (MacMillan, New York, 1964). 6. Z. JANKO: A new finite simple group with Abelian Sylow 2-subgroups and its characterization. J. of Algebra, 3 (1966), 147-186. A programme for the calculation of characters and representations of’nite groups C. BROTTAND J.NEUB&ER 1. Introdnction. The programme described in this paper is part of a system of programmes for the investigation of finite groups. Other parts of this system are described in [3], [4]. ,‘I ._ The programme avoids numerical calculations as far as possible. Instead properties of the given group which are available from other programmes have been used to construct characters and representations by the process of induction. Only when this process does not yield all the required information does the programme use numerical methods. The programme has been started as a “Diplomarbeit” [l]. We are grateful to the Deutsche Forschungsgemeinschaft for financial support and to Prof. K. H. Weise for opportunities given to us at the “Rechenzentrum der Universitgt Kiel”. We would like to thank Mr. V. Felsch for valuable -help in connecting this programme with the one described in [3]. 1.1. Notations. All groups considered are finite, they are denoted by CH, . . ..(gl. . . . . gJ is the subgroup generated by the elements gr, . . ., ge E G; G’ = (x-’ y-lxyl X, y E G) is the commutator subgroup of G; Cl, . . . . C, are the classes of elements conjugate in G; Cr = {I}, hi is the number of elements in Ci. The structure constants cijk are defined by (1.1.1) Z is the ring of integers, Z, the field of integers module a prime p, C the complex field, Mr, . . . , M, the set of all absolutely irreducible CG-modules with dimensions dl, . . . , d,. $J> is the character belonging to Mj, djJ its value on Ci. All representations considered are C-representations, so irreducible always means absolutely irreducible. 2. Available programmes for the investigation of finite groups. Our programme X for the calculation of characters and representations of a given finite group G makes use of the data determined by a programme @ for the investigation of the lattice of subgroups and of certain other properties of G. The input for the system of programmes consisting of @ and X is a set of generators of G given in one of the following ways: 101
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COMPUTATIONAL PROBLEMS IN ABSTRACT
Page 3 and 4: vi Contents E. KRAUSE and K. WESTON
Page 5 and 6: X Preface I am indebted to the auth
Page 7 and 8: 4 J. Neubiiser 2.3.5. Added in proo
Page 9 and 10: 8 J. Neubiiser Investigations of gr
Page 11 and 12: 12 J. Neubiiser For 1 es 4 r, ws is
Page 13 and 14: 16 J. Neubiiser Investigations of g
Page 23 and 24: Some examples using coset enumerati
Page 25 and 26: 40 C. M. Campbell Some examples usi
Page 27 and 28: 44 N. S. Mendelsohn for example, in
Page 29 and 30: 48 H. Jiirgensen Calculation with e
Page 35 and 36: 60 V. Fe&h and J. Neubiiser 2. The
Page 37 and 38: 64 L. Gerhards and E. Altmann Autom
Page 39 and 40: 68 L. Gerhards and E. Altmann Autom
Page 41 and 42: 72 L. Gerhards and E. Altmann ni E
Page 43 and 44: 76 W. Lindenberg and L. Gerhards Se
Page 45 and 46: 80 W. Lindenberg and L. Gerhards Se
Page 47 and 48: 84 K. Ferber and H. Jiirgensen The
Page 49 and 50: 7 The construction of the character
Page 51 and 52: 92 John McKay defining multiplicati
Page 53: 96 John McKay Construction of chara
Page 57 and 58: 104 C. Brott and J. Neubiiser irred
Page 59 and 60: 108 C. Brott and J. Neubiiser eleme
Page 61 and 62: 112 J. S. Frame The characters of t
Page 63 and 64: 116 J. S. Frame 3. The decompositio
Page 69 and 70: TABLE 5. The Z, character block for
Page 71 and 72: 132 R. Biilow and J. Neubiiser Deri
Page 73 and 74: A search for simple groups of order
Page 75 and 76: 140 Marshall Hall Jr. Simple groups
Page 79 and 80: 148 Marshall Hall Jr. otherwise no
Page 81 and 82: 152 Marshall Hall Jr. easily found
Page 85 and 86: 160 Marshall Hail Jr. This leads to
Page 87 and 88: 164 Marshall Hall Jr. b= (OO)(Ol, 0
Page 89 and 90: 168 Marshall Hall Jr. 11. R. BRAUER
Page 91 and 92: 172 Charles C. Sims THEOREM 2.3. Th
Page 93 and 94: 176 Charles C. Sims has a set of im
Page 95 and 96: 180 Degrc - No. Order t N (-63 Gene
Page 97 and 98: An algorithm related to the restric
Page 99 and 100: A module-theoretic computation rela
Page 101 and 102: 192 A. L. Tritter expressed in the
Page 103 and 104: 196 A. L. Tritter a question is mos
Page 105 and 106:
200 John J. Cannon stack. The Cayle
Page 107 and 108:
, I The computation of irreducible
Page 109 and 110:
208 P. G. Ruud and R. Keown product
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212 P. G. Ruud and R. Keown TX wher
Page 113 and 114:
216 P. G. Ruud and R. Keown 4. L. G
Page 115 and 116:
220 N. S. Mendelsohn where it is kn
Page 117 and 118:
224 Robert J. Plemmons such class [
Page 119 and 120:
228 Robert J. Plemmons ! TOTALS Sem
Page 121 and 122:
232 Takayuki Tamura 8” = 0B if an
Page 123 and 124:
236 Takayuki Tamura Case ,Q = {e).
Page 125 and 126:
240 Takayuki Tamura The calculation
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244 Takayuki Tamura We calculate46
Page 129 and 130:
248 Takayuki Tamura Immediately we
Page 131 and 132:
252 Takayuki Tamura Let U be a subs
Page 133 and 134:
256 Takayuki Tamura TABLE 8. AI1 Se
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I The author and R. Dickinson have
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264 Donald E. Knuth and Peter B. Be
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Page 155 and 156:
300 Lowell J. Paige Non-associative
Page 157 and 158:
304 Lowell J. Paige We may lineariz
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308 T(n) U(n) 0) [VI R-4 C. M. Glen
Page 161 and 162:
312 C. M. Glennie where in each cas
Page 163 and 164:
316 A. D. Keedwell of the elements
Page 165 and 166:
A projective coqfiguration J. W. P.
Page 167 and 168:
The uses of computers in Galois the
Page 169 and 170:
328 W. D. Maurer mod p; for a facto
Page 171 and 172:
332 J. H. Conway Enumeration of kno
Page 173 and 174:
J. H. Conway Enumeration of knots a
Page 175 and 176:
340 J. H. Conway -thus our symmetry
Page 177 and 178:
344 J. H. Conway 2 string links to
Page 179 and 180:
348 J. H. Conway 3 and 4 string lin
Page 181 and 182:
352 Non-alternating J. H. Conway L
Page 183 and 184:
356 J. H. Conway Alternating 11 cro
Page 185 and 186:
H. F. Trotter Computations in knot
Page 187 and 188:
364 H. F. Trotter to a. Schubert [1
Page 189 and 190:
368 Shen Lin sequence of squares, t
Page 191 and 192:
372 Harvey Cohn We also consider GZ
Page 193 and 194:
376 Harvey Cohn In the case of Fig.
Page 195 and 196:
380 Harvey Cohn always dictated by
Page 197 and 198:
384 Hans Zassenhaus A real root cal
Page 199 and 200:
388 Hans Zassenhaus the elements A,
Page 201 and 202:
392 Hans Zassenhaus It is clear fro
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396 R. E. Kalman Invariant factors
Page 205 and 206:
400 List of participants DR. J. J.
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