COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

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The application of computers to research in non-associative algebras LOWELL J. PAIGE 1. Introduction. The number of papers presented here at this conference indicate a wide area of computer applications in algebraic research; group theory, algebraic topology, galois theory, knot theory, crystallography and error correcting codes. I wish to confine my remarks to less specific details, and I will indicate where the computer has been used in my own research and in the work of others involved with non-associative systems. It seems to me that the computer can and does play different roles in algebraic research. I would classify the potential of computer assisted research in the following manner: (A) The computer can provide immediate access to many examples of any algebraic structure so that reasonable conjectures may be formulated for more general (possibly machine free) investigation. (B) The computer can be used for a search for counter-examples of a general conjecture. (C) The computer can be used to provide the “proof” required in a mathematical argument. In the next section, I shall attempt to indicate by means of various examples where the computer has led to success and failure in the categories listed above. Finally, I would like to suggest in the field of Jordan algebras the possibility of computer assistance to attack the general problem of identities in special Jordan algebras. 2. Examples of computer assisted research. My own introduction to computer assistance in research arose in an investigation of complete mappings of finite groups. This problem stems from an early attempt to construct a finite projective plane by means of homogeneous coordinates from a neofield, and the problem for groups may be stated briefly as follows: Let G be a finite group (written multiplicatively) and let 9 be a bijection of G. For what groups G is the mapping 7 : x - x-O(x) a bijection of G? A complete solution for this problem in the case that G is abelian was obtained in 1947 [l]. I obtained some fragmentary results for non-abelian 299
300 Lowell J. Paige Non-associative algebras 301 groups in 1951 [2] and Professor M. Hall generalized the results for the abelian case in 1952 [3]. It was at this time (1953) that I sought help from the computer and obtained a detailed analysis of complete mappings for groups of small order. The memory capacity of SWAC at that time prevented any large-scale analysis, but the results were of such a nature that Professor Hall and I reconsidered the problem. We gave a complete solution to the problem for solvable groups and stated the following conjecture in 1955 [4]: CONJECTURE: A jinite group G whose Sylow 2-subgroup is non-cyclic possesses a complete mapping. This conjecture has never been verified nor has a counterexample been found. Computers could certainly provide more evidence, but the solution of the problem has rather dubious applications to the original problem of projective planes. There is, however, an interesting footnote to the conjecture. I felt that I could provide a solution if the following published problem of 1954 were true: PROBLEM. Let G be a finite group and Sz a Sylow 2-subgroup. In the coset decomposition of G by Sz, does there exist an element of odd order in each coset ? The problem remained unsolved until Professor John Thompson provided a counterexample in 1965. His example was the group of 2 x 2 unimodular matrices over the Galois Field GF(53). It is easy to see that the Sylow 2-subgroup of Thompson’s example is non-cyclic, and there is reason to suppose that this example might provide a counter-example to our original conjecture; however, the order of this group (148,824) makes it seem unlikely that even today’s computers would be capable of providing the answer. My experience with computers and 10 x 10 orthogonal lattice squares was not a particularly successful venture. In 1958, I wrote, “consequently, the total time necessary to do an exhaustive search for latin squares orthogonal to our example would be approximately 4.8 X 1On machine hours”. Perhaps the time computation was correct but we are all well aware that the counter-example to Euler’s conjecture was provided the next year. An example of a computer-provided counter-example to another of Euler’s conjectures occurred last January when L. J. Lander and T. R. Parkin published the following numerical relation [5] : 275+-845+1105+133~ = 1445. Let me turn now to an example from loop theory for a more favorable experience with a “machine suggested” conjecture. First, a brief review of the pertinent loop theory. Two loops (G, .) and (H, X ) are said to be isotopic if there exists a triple of bijections (cc, j, y) of G to H such that WX(YB) = (X-Y& for all x, y of G . Professor R. H. Bruck raised the following problem in 1958 [6]: Find necessary and sufficient conditions upon a loop G in order that every loop isotopic to G is isomorphic to G. The answer to this problem, as was pointed out by Bruck, would have interesting interpretations to projective planes. It is well known that a group G has the property that it is isomorphic to all of its loop isotopes. A student at the University of Wisconsin, working under the direction of Professor H. Schneider, examined all loops of order 7 on a computer and discovered that the only loop isomorphic to all of its loop isotopes was the cyclic group. He has proved subsequently that any loop of prime order satisfying the property that it was isomorphic to all of its loop isotopes must be the cyclic group. Moreover, I understand that he is now considering a generalization of this result to loops of prime power order. Another example which comes to mind involves a question raised by Professor T. A. Springer [7] concerning elements in a Chevalley group; specifically, “Is the centralizer, G,, of a regular unipotent element .Y, an abelian group ?” Dr. B. Lou, working under the direction of Professor Steinberg at the University of California, Los Angeles, has given the answer to this question in those cases left open by Springer, and a considerable portion of her work in the associated Lie algebras was done on a computer. It would be a matter of serious negligence in surveying the applications of computers to non-associative systems if one were not to mention the work of Professor Kleinfeld [8] on Veblen-Wedderburn systems with 16 elements, or that of Professor R. Walker [9] in extending these results to a listing of finite division algebras with 32 elements. Finally, the work of Professor D. Knuth [10] in providing new finite division algebras is an excellent example of computer assisted research. 3. Jordan algebra identities. The problem of determining identities satisfied by the elements of a non-associative algebra is one area in which the computer could be expected to make research contributions. For example, Professor R. Brown of the University of California has discovered an identity in one variable for certain algebras arising in his investigation of possible representations of the Lie group associated with ET. Professor M. Osborne of the University of Wisconsin has published all of the possible identities of degree 4 or less for commutative algebras. These were done without the aid of a computer. However, Professor Koecker of Munich has sought computer assistance in determining all possible identities of a non-associative algebra where the degree of the identity is restricted to degree 6 or less. The early estimates of the machine time required for
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COMPUTATIONAL PROBLEMS IN ABSTRACT
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vi Contents E. KRAUSE and K. WESTON
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X Preface I am indebted to the auth
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4 J. Neubiiser 2.3.5. Added in proo
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8 J. Neubiiser Investigations of gr
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12 J. Neubiiser For 1 es 4 r, ws is
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16 J. Neubiiser Investigations of g
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Some examples using coset enumerati
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40 C. M. Campbell Some examples usi
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44 N. S. Mendelsohn for example, in
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48 H. Jiirgensen Calculation with e
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60 V. Fe&h and J. Neubiiser 2. The
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64 L. Gerhards and E. Altmann Autom
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68 L. Gerhards and E. Altmann Autom
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72 L. Gerhards and E. Altmann ni E
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76 W. Lindenberg and L. Gerhards Se
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80 W. Lindenberg and L. Gerhards Se
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84 K. Ferber and H. Jiirgensen The
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7 The construction of the character
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92 John McKay defining multiplicati
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96 John McKay Construction of chara
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100 John McKay In the table, c indi
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104 C. Brott and J. Neubiiser irred
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108 C. Brott and J. Neubiiser eleme
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112 J. S. Frame The characters of t
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116 J. S. Frame 3. The decompositio
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TABLE 5. The Z, character block for
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132 R. Biilow and J. Neubiiser Deri
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A search for simple groups of order
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140 Marshall Hall Jr. Simple groups
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148 Marshall Hall Jr. otherwise no
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152 Marshall Hall Jr. easily found
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160 Marshall Hail Jr. This leads to
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164 Marshall Hall Jr. b= (OO)(Ol, 0
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168 Marshall Hall Jr. 11. R. BRAUER
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172 Charles C. Sims THEOREM 2.3. Th
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176 Charles C. Sims has a set of im
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180 Degrc - No. Order t N (-63 Gene
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An algorithm related to the restric
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A module-theoretic computation rela
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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
Page 111 and 112: 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
Page 127 and 128: 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
Page 135 and 136: I The author and R. Dickinson have
Page 137 and 138: 264 Donald E. Knuth and Peter B. Be
Page 153: 296 Donald E. Knuth and Peter B. Be
Page 157 and 158: 304 Lowell J. Paige We may lineariz
Page 159 and 160: 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
Page 203 and 204: 396 R. E. Kalman Invariant factors
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400 List of participants DR. J. J.
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