COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

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198 A. L. Tritter is found to be “yes”, in case it is found to be “no”, in which of two modes a continuously monitored visual display is to show progress, and finally something slightly peculiar. Suppose that an input discrepancy is found on, say, a B-C pass. A decision is taken to “back off” and run A -+ B again. Now, if this decision is implemented immediately, no new problem arises. But, suppose as well that we interrupt at this point; then, when we go back onto the machine to re-run A-B, there is no record anywhere except in our own minds that it is not possible to back off from this pass (as C is not the predecessor of A here-it has been altered on the abortive B-C pass). The last switch is used to inform the program that thejirst magnetic tape pass of the present machine run is one from which no back-off is possible; it is, of course, used on the very first run, as well as in the more complex situation described above. One more word about the triangularization. It has been necessary, for efficiency’s sake, to economize as much as possible on the total time for which the program occupies Atlas, and this has meant developing a new triangularization algorithm. This algorithm will be discussed elsewhere, rather than here. It is a natural modification of existing methods, simply taking maximal advantage of the fact that the matrix is over a finite field while, at the same time, using the space outside the upper right triangle of the matrix, as it is progressively vacated by the triangularization, as the place to record all the “pedigree” data (there is exactly the necessary amount of space). 8. Present status. The matrix-generating program is written, debugged, and run. The triangularization algorithm has been tested by hand, and is correct. The triangularization program is written and undergoing debugging. REFERENCES 1. C. R. B. W RIGHT: On the nilpotency class of a group of exponent four. Pac. J. Maths. 11 (1961), 387-394. 2. Unpublished private communication. 3. S. M. JOHNSON: Generation of permutations by adjacent transposition. Maths. Comp. 17 (1963), 282-285. 4. J. ARMIGER TROLLOPE: Grandsire: The Jasper Snowdon Change-ringing Series, pp. 51, 122 (Whitehead & Miller, 1948). 5. DOROTHY L. SAYERS: The Nine Tailors (Victor Gollancz, 1934, many reprints). Some combinatorial and symbol manipulation programs in group theory JOHN J. CANNON Introduction. Over the past two years computers have been used to carry out a number of large calculations, of both a numerical and nonnumerical nature, arising out of research in group theory at Sydney. These problems include : (i) construction of subgroup lattices; (ii) investigation of positive quadratic forms; (iii) determination of the groups of order p6, p > 2; (iv) construction of counter-examples to Hughes’ conjecture in group theory. Only (i), (iii) and (iv) will be discussed here. All programs described here have been written in the English Electric KDF9 Assembly Language, USERCODE. Subgroup lattices. A program has been written which determines the generators and relations of all the subgroups of a finite soluble group. The program finds the subgroups using the same method as Neubiiser [l] with the difference that as a subgroup of order d is found the generators and relations of all possible groups of order d are checked through to find a set of generators and relations for the subgroup. As the program is restricted to groups of order less than 400 by machine considerations, one only needs to know all the possible groups of order less than 200, and most of these are known. The program reads the generators and relations of the given group and, using coset enumeration, finds a faithful permutation representation (possibly the regular representation). The permutation representation is used to determine the elements of the group and to find its Cayley table, and is then discarded. At the same time as the group elements are being found as permutations they are also found as words in the original abstract generators. These n words are stored in an n-word stack so that instead of using either the abstract word or its permutation representation, one uses the number indicating the position of the group element in this CPA 14 199
200 John J. Cannon stack. The Cayley table may then be stored compactly as a Latin square with several entries stored in a single machine word. In this manner the multiplication tables of groups of order up to 380 may be kept in-the core of a 32K machine. Only when group elements are being output is reference made to the stack of group words. The user has the option of outputting either the generators and relations of each subgroup or the elements of the subgroup or both. So far the program has been run successfully for a number of small groups (all of order less than 100). Before larger groups can be run more sets of generators and relations have to be checked and included in the program. Investigation of groups of ordeq#,p =- 2. R. James has been enumerating the p-groups of order p6, p > 2, checking the work of Easterfield, by the method of isoclinism. Isoclinism [2, 3] splits the possible sets of generators and relations into classes so that all members of a class have the same commutator relations. The problem then is to find all the non-isomorphic groups within each isoclinism class. We may suppose that isomorphism is an automorphism and apply a general automorphism to a general set of generators and relations (remembering that commutator relations are invariant within an isoclinism class). This will give rise to a set of relations on the integers mod n, for some n (obtained by equating indices of each generator before and after the automorphism). For p-groups, n = pr for some r. These relations can be expressed as an equivalence relation on a set of matrices over GF(p). The equivalence classes of these matrices give the non-isomorphic groups of this isoclinism class. Difficulties arise in the determination of a set of equivalence class representatives for general p and it is to this problem that a computer has been applied. Using a computer it is simple to calculate the equivalence classes and to select a suitable equivalence class representative (an element which will give the corresponding generators and relations in the simplest possible form) for each class, for the first few primes. It is then usually easy to write down a set of equivalence class representatives for general p. The groups of order p5 (p > 2) were checked by this method and the groups of orderpe (p * 2) are being found. Hughes’ conjecture and commutator calculations. Consider a group G of order p” where p is a prime. Take the subgroup H of G generated by the elements of G having order greater than p. We suppose that G has some elements whose orders are greater than p. Then Hughes’ conjecture is that H is of index 1 or p in C. It is true for p = 2, 3. G. E. Wall has shown that the conjecture is false for p if a certain expression E is zero in a Lie algebra generated by two elements of nilpotency class 2p- 1, satisfying the (p - 1)th Engel condition, over GF(p). Combinatorial and symbol manipulation programs 201 Hand calculations carried out by Wall show that the conjecture is false for p = 5. As the calculations are extremely long and tedious, it was decided to develop programs to verify the calculations for p = 5, and to carry them out for p = 7 and 11. The problem will be considered in four parts: (1) Determination of a basis for the two-generator free Lie algebra of nilpotency class 2p - 1. (2) Determination of a basis of the algebra in (1) with the (p- 1)th Engel condition imposed. (3) Calculation of the expression E. (4) Expression of E in terms of the basis elements found in (2). These will now be considered in turn. (1) Let 5 and 7 denote the two generators of the algebra. We define the weight of an arbitrary element of the algebra as follows: The elements 5 and 7 are of weight one. If A = (P, Q) is an element of the algebra, then weight A = weight P+weight Q. Basic commutators are next defined together with an ordering (0 on them. The elements 5 and 7 are basic. Under the ordering all basic commutators of weight w come after those of weight w- 1. Ordering is arbitrary among basic commutators of weight w, but once an ordering is chosen it must be adhered to. A commutator C = (A, B), A = (P, Q), where A and B are basic, is basic if A =- B and B 3 Q. The elements of weight w form a subspace of the algebra and a basis of this subspace is provided by the basic commutators of weight w [4]. Given an element of the algebra, (A, B), where A = Z&C’,, B = ZpjC”, Ci, C’ basic, we describe a collection process [5]. ’ (i) Put (A, B) = zlipj (Ci, Cj)* (ii) If Ci and C’ are basic, put ta) (Ci9 cj) = O if cg.Z Cip tb) tci9 cj) = - (Cj, Ci) if Ci ( Cj, (C) (Ci, Cj) = (Ci, Cj) if Ci > C* J’ (iii) If Ci * Cj are basic and Ci = (Pi, Qi), put (a) (Ci, C+‘> = ((pi, Qi>, C’I if Cj * Qi, Cb) Cci9ccj> = -((Qi, cj), pi) +((pi, cj), Qi> if Cj < Qi* (iv) Return to (i) and repeat the process until (A, B) is expressed as a linear combination of basic commutators. A program has been developed which calculates the basic commutators
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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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Page 35 and 36:
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
Page 53 and 54: 96 John McKay Construction of chara
Page 55 and 56: 100 John McKay In the table, c indi
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: 196 A. L. Tritter a question is mos
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
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300 Lowell J. Paige Non-associative
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304 Lowell J. Paige We may lineariz
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308 T(n) U(n) 0) [VI R-4 C. M. Glen
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312 C. M. Glennie where in each cas
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316 A. D. Keedwell of the elements
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A projective coqfiguration J. W. P.
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The uses of computers in Galois the
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328 W. D. Maurer mod p; for a facto
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332 J. H. Conway Enumeration of kno
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J. H. Conway Enumeration of knots a
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340 J. H. Conway -thus our symmetry
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344 J. H. Conway 2 string links to
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348 J. H. Conway 3 and 4 string lin
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352 Non-alternating J. H. Conway L
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356 J. H. Conway Alternating 11 cro
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H. F. Trotter Computations in knot
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364 H. F. Trotter to a. Schubert [1
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368 Shen Lin sequence of squares, t
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372 Harvey Cohn We also consider GZ
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376 Harvey Cohn In the case of Fig.
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380 Harvey Cohn always dictated by
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384 Hans Zassenhaus A real root cal
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388 Hans Zassenhaus the elements A,
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392 Hans Zassenhaus It is clear fro
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396 R. E. Kalman Invariant factors
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400 List of participants DR. J. J.
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COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

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