COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.
COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA. COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.
50 H. Jiirgensen Calculation with elements of a finite group 51 4. Multiplication programme. A machine programme which strictly follows this algorithm will be rather slow; the reasons are that it is recursive, and that the input data, i.e. the words g, and the integers vj and n, which are constants throughout all calculations with the elements of a given group, have to be looked up for every multiplication again and again. We could naturally save some computing machine time if a special multiplication programme was written for every AG-system. For a trained programmer this might take less time than doing all the group calculation by hand. As had been realized for groups whose orders are powers of 2 by Neubiiser, and with a different sort of defining systems for finite soluble groups by Lindenberg, a compromise would be to write a programming programme C, which will generate the multiplication programme M needed according to the special AG-system given. I shall give a rough sketch of C in terms of M. C has been running at Kiel since 1965. Mis generated in two steps. First a multiplication programme M* and an inverting programme ZNV are generated, for which just the input data, i.e. the AG-system given, are used. Then M itself is generated as an improved version of M* in such a way that parts of M* are replaced by new ones according to further relations which will be computed by M*. M and M* both consist of a main programme H, the structure of which is the same in M and M*, and a set of subroutines P(i, j, k). 4.1. The programme H (see flow chart). The normed elements d;a>$ . . . a: E G* are uniquely represented by the integers El + i ci. ‘d 2lfentier (log&-l)) i-2 ( j=l 1 With 27 binary digits per machine word of the electronic computer used by us (Electrologica Xl) the group elements can normally be stored in one word each, when this representation is used.+ In the beginning of H the exponents Ei of b and ai of c (i = I(l)n) are isolated and stored to separate machine words. tThe number of binary digits needed for the elements of a group G is n + 2 entier (log,(y,- 1)) c n+,zI entier (log, v,) =5 n + entier (log, fi ~8‘) i=l The representation which will need the fewest binary digits is %+~2(%*;~Y+ l-.1 = n+entier (log, ICI). for a binary machine; however, isolating the Ei will generally take more time, because divisions instead of shifting operations will be needed for the elements stored like this. Then for i = l(l)n the element wi = aja+a$:;. . . a? is normed ei times in such a way that the machine words, where the exponents 6i are stored, will afterwards contain the exponents of the result. For i L- l(l)k and k = l( 1)n we define the natural number O(i, k) by: gi, e
52 H. Jiirgensen Calculation with elements qf aJinite group 53 4.2. The subroutines P(i, j, vi> (1 =~i-=j=~n). These subroutines just consist of a “go-to-instruction”, which will lead to the subroutine P(i, j, Sj) for +O and to P(i, e(i, j-l),~8Ci,i-1,) for 6j = 0, according to the actual value of Sj (see flow chart). 4.3. The subroutines P(i, j, k) (I=~i (1 6 j =S n). The element a&@.-~ . . . CZ? will be normed. P(j, j, vj) will call further subroutines according to ga, if Sj+ 1 = yj (see flow chart).? 4.5. The inverting programme (see flow chart). When M* is finished an inverting programme INV will be generated. The word w = a,““af,y--; . . . q c G* is inverted by successively norming the following words : apl-%e = w1 aO,a~~-;h-%-l~, = wz . azaiml . . . a,a, 0 I%bIW = w, = w-1 II-1 (a) t All those subroutines P(i, j, k) will be generated for M*, for which l-=i=jen and k=yJ, or (b) 1 =si-=j=sn and lekecy, and gtj+e, or (C) i = j = 1 and O(l, n) + 1 and k = vl.
- Page 1 and 2: 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: 48 H. Jiirgensen Calculation with e
- Page 33 and 34: 56 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 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 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 77 and 78: 144 Marshall Hall Jr. Simple groups
- Page 79 and 80: 148 Marshall Hall Jr. otherwise no
52<br />
H. Jiirgensen<br />
Calculation with elements qf aJinite group 53<br />
4.2. The subroutines P(i, j, vi> (1 =~i-=j=~n). These subroutines just consist<br />
of a “go-to-instruction”, which will lead to the subroutine P(i, j, Sj)<br />
for +O and to P(i, e(i, j-l),~8Ci,i-1,) for 6j = 0, according to the<br />
actual value of Sj (see flow chart).<br />
4.3. The subroutines P(i, j, k) (I=~i (1 6 j =S n). The element a&@.-~ . . .<br />
CZ? will be normed. P(j, j, vj) will call further subroutines according to<br />
ga, if Sj+ 1 = yj (see flow chart).?<br />
4.5. The inverting programme (see flow chart). When M* is finished an<br />
inverting programme <strong>IN</strong>V will be generated.<br />
The word w = a,““af,y--; . . . q c G* is inverted by successively norming the<br />
following words :<br />
apl-%e = w1<br />
aO,a~~-;h-%-l~, = wz<br />
.<br />
azaiml . . . a,a, 0 I%bIW = w, = w-1<br />
II-1<br />
(a)<br />
t All those subroutines P(i, j, k) will be generated for M*, for which<br />
l-=i=jen and k=yJ, or (b) 1 =si-=j=sn and lekecy, and gtj+e, or<br />
(C) i = j = 1 and O(l, n) + 1 and k = vl.
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