COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

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218 N. S. Mendelsohn few values of yt. Here are some of the results. GI = I; Gz = I; G3 = LF(2, 7), with a faithful representation A * (26) (34), B - (23) (4567), c * (12) (45). It is to be noted that Gs is the collineation group of the Fano projective plane which contains seven points. Gq = LF(2, 7) with a faithful representation (using subscripted letters) Al * (26) (34), B1 -+ (23) (4567), Cl + (12) (67). An isomorphism between G4 and GS is given by the mapping Al-A, BlwB, Cl++B-2CB2. Gs is a group of order 1080. A faithful representation of degree 45 which is equivalent to the representation given by the permutation of the subsets of the subgroup generated by A and B under right multiplication is given by A + (1) (5) (7) (12) (35) (36) (41) (44) (45) (236) (334) (8, 9) (1% 11) (13, 14) (15, 24) (16, 23) (17, 18) (19, 34) (20, 21) (22, 25) (26, 30) (27, 28) (29, 33) (31, 32) (37,43) (38, 39) (40,42), B --f (1) (44) (45) (2, 3) (32, 43) (39, 40) (4, 5, 6, 7) (8, 16, 17, 22) (9, 29, 14, 15) (10, 13, 28, 21) (11, 34, 23, 24) (12, 38,41,42) (18, 19, 20, 30) (25, 26, 27, 33) (31, 36, 37, 35), c + (3) (9) (13) (16) (19) (24) (25) (32) (40) (1, 2) (4, 10) (5, 17) (6, 8) (7, 28) (11, 12) (14, 30) (15, 37) (18, 31)(20, 35)(21,22)(23,38) (26, 41) (27, 34) (29, 36) (33, 42) (39, 45) (43, 44). Marshall Hall pointed out to the author that this representation is imprimitive with 1,44, 45 a set of imprimitivity. By considering the representation obtained by permuting the sets of imprimitivity one obtains a factor group of GS of order 360, and hence Gs has a normal subgroup of order 3. A faithful representation of this factor group by permutations on 15 symbols is given by A - (1) (5) (7) (226) (3,4) (8, 9) (10, 11) (12, 15) (13, 14), B - (1) (2, 3) (4, 5, 6, 7) (8, 9, 11, 14) (10, 13, 12, 15), C -+ (3) (9) (13) (1, 2) (4, 10) (5, 11) (6, 8) (7, 12) (14, 15). Examples of man-machine interaction 219 This factor group is not A5 since it has S4 as a subgroup while A5 contains no elements of order 4. G,. The group Gs is of infinite order. In this case we find a homomorphism of GC onto SL(3, Z), where the latter is understood to be the set of all non-singular matrices of order 3 and determinant 1 and whose entries are integers. The following mapping exhibits the homomorphism explicitly : A direct verification shows that the defining relations are satisfied by these matrices. On the other hand the three matrices generate SL(3, Z) as is seen by the mappings X = C(AC)3B2AB2C + AXA B2AB2XAB2AB2A --t
220 N. S. Mendelsohn where it is known that the six matrices appearing on the right generate SL(3, Z). The following geometrical corollary then follows. The group generated by all relations of a finite Desarguesian plane which is generated by four points is a homomorphic image of the collineation group of the free plane generated by four points. Example 2. Complete latin squares. A complete latin square of order n is an array in which every row and every column is a permutation of n symbols and such that every ordered pair of symbols appears as a consecutive pair exactly once in the rows and once in the columns. For example A B C D C A D B B D A C D C B A is a complete latin square of order 4. In [2] B. Gordon has shown that complete latin squares exist for every even order and in [1] E. N. Gilbert has given a number of special constructions all for even order. It is known that for n = 3, 5, 7, 9 no complete latin square exists and it had been conjectured that none exists for any odd order. An exhaustive search by machine is quite impractical for n Z= 11 so that a specialized search based on an incomplete mathematical theory may be of use. Following Gordon we look for a solution in which the square is the multiplication table for a group. The problem is then reduced to the following. Let 81, g2, . . ., g,, be the distinct elements of a group. Is it possible to arrange them so that the elements gl, gF1gZ, g;lgs, . . ., g,=‘,g,, are all distinct ? If the group is of odd order and Abelian it is known to be impossible for such an arrangement to exist. Hence we look to groups which are non-abelian. The smallest order of a non-abelian group of odd order is 21. However, a search through the 21! permutations of the elements of the group is still impractical. The compromise used was to start the arrangement of the elements of the group by hand until one gets stuck (usually after 16 to 18 elements). When this is finished the arrangement was put into the machine with a back-tracking program to try to alter and complete the arrangement. This proved eminently successful. On p. 221 is an example of one of the latin squares of order 21. Example 3. Commutators. There are a number of combinatorial problems in which it is important to know whether or not an element of a commutator subgroup is a commutator. More generally the following question is of interest. Given an element A of the commutator subgroup, Examples of man-machine interaction An example of a latin square of order 21 without repeated digraphs (both rows and columns) A B C D E F G H I J K L M N O P Q R S T U K A P I S M T R Q F B E J H C O D N L U G O M L K B R F D S Q H N E C T J P G U A I P F S A K H J Q E I N R L O G M C U T B D E D A J C P S L H B T I G F Q N U K M R O R C F L G S K T O N J A D I E U B P H Q M C J E B A N M I L D R H S P U F O T G K Q F N G C R I D O A L P T H U M S E Q K J B LQBMOCESNKGDUJ’RTAFHP M H T P N Q I C K E O U R G J L S D B F A S l K F P O L E R A U Q T M D H G B J N C Q G I T D K P J M U L F C A R B N S O E H T E N H M G R K U P D C B L A Q J F I O S N P M E T L B U C H F K I Q S G A O R D J B K O Q L J U N D M A S F R P C I H E G T G L R N F U H A T C Q O K S B I M J D P E H O J S U E A G P R M B Q D L T K C N I F DTQUIBOFJGEMPKh’AHLCSR JRUOHDQPBsCGNTFELIAMK I U D G Q A C M F T S J O B H K R E P L N U S H R J T N B G O I P A E K D F M Q C L find the minimum value k such that A is expressible as a product of k commutators. This problem, in general, is known to be unsolvable. However, for free groups it is solvable but the solution is by no means trivial. The author has studied the problem of a machine program for making the computations. To test the possible efficiency of the program the author carried out hand calculations which would imitate the machine’s behaviour. Naturally, relations were generated in a random order, but a number of them proved to be very interesting and one of these led to an interesting theorem. Here are some of the results turned out at random, all referring to free groups. (a) In a free group a word of length six or less which lies in the commutator subgroup is a commutator, e.g. a-lb-lc-labc = (ba, bc). 221
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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
Page 63 and 64: 116 J. S. Frame 3. The decompositio
Page 65 and 66: 120 J. S. Frame The characters of t
Page 67 and 68: 124 J. S. Frame The characters of t
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
Page 111 and 112: 212 P. G. Ruud and R. Keown TX wher
Page 113: 216 P. G. Ruud and R. Keown 4. L. G
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 155 and 156: 300 Lowell J. Paige Non-associative
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
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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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