COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.
COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA. COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.
54 H. Jiirgensen Calculation with elements of a finite group 55 It is important here that the multiplication algorithm is defined for seminormed words as left factors. For computing jail (i = l(l)n) M* is used. 4.6. M developedfrom M*. What proved to be most time-consuming% that with an AG-system given the number of steps to be taken for norming the word aia%&sj-l I 3 1” .a> will in general increase rather fast, when Sj is increased. As a remedy further normed words h, = [ai, a:] (1 e i -= j 5 n; 2 == k-= vi> are introduced, if there exist i and j (1 e i-=j=sn) such that vj + 2 and gii + e. For computing these normed words M* is used. In M* norming a,u,dra$:;. . .a? for i-=j and 6j>l is based on the equation aiais,& 8,-l 6,-I a1‘I = ajaigija, . . . C7j-l . . . a$. The subroutines P(i, j, k) of M* for i
56 H. Jiirgensen Calculation with elements of a finite group 57 of commutators by which it may be defined.? Such systems (P-systems) exist as defining ones for every finite soluble group and even for some finite groups which are not soluble. If opj) = 1 (i = l(l)j- 1; j = 2(l)n), i.e. the group is soluble, there exists a well-defined finite multiplication algorithm. (d) Some non-soluble finite groups may be calculated, when an “extended P-system” is given. As far as P-systems are concerned, the extended ones seem to be the “weakest” with a well-defined finite multiplication algorithm existing. A new version of the programming programme, which is just being written, will allow the input data, i.e. the defining system, to be a mixture of AG-, K-, P-, and extended P-systems, and the words to be not necessarily normed ones. REFERENCES 1. W. LINDENBERG: ijber eine Darstelhmg von Gruppenelementen in digitalen Rechenautomaten. Num. Math. 4 (1962), 151-153. 2. W. LINDENBERG: Die Struktur eines Obersetzungsprogrammes zur Multiplikation von Gruppenelementen in digitalen Rechenautomaten. Mitt. Rh- W. Inst. Znstr. Math. Bonn 2 (1963), l-38. 3. J. NEUB~~SER: Bestimmung der Untergruppenverblnde endlicher p-Gruppen auf einer programmgesteuerten elektronischen Dualmaschine. Num. Math. 3 (1961), 271-278. 4. See the papers by C. BROTT, R. BULOW, K. FERBER, V. FELSCH and J. NEUB~SER in these Proceedings. t at = ai= ai = aa = e, ala, = a,a,, ala3 = a3a2, ala, = afa,a,al, a,u, = a,a,a,, a2a4 = a,4a,, u3a4 = a4a$a2u1.
- 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 and 30: 48 H. Jiirgensen Calculation with e
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- 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
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- 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
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56 H. Jiirgensen Calculation with elements of a finite group 57<br />
of commutators by which it may be defined.? Such systems (P-systems)<br />
exist as defining ones for every finite soluble group and even for some finite<br />
groups which are not soluble. If opj) = 1 (i = l(l)j- 1; j = 2(l)n), i.e. the<br />
group is soluble, there exists a well-defined finite multiplication algorithm.<br />
(d) Some non-soluble finite groups may be calculated, when an “extended<br />
P-system” is given. As far as P-systems are concerned, the extended ones<br />
seem to be the “weakest” with a well-defined finite multiplication algorithm<br />
existing.<br />
A new version of the programming programme, which is just being written,<br />
will allow the input data, i.e. the defining system, to be a mixture of<br />
AG-, K-, P-, and extended P-systems, and the words to be not necessarily<br />
normed ones.<br />
REFERENCES<br />
1. W. L<strong>IN</strong>DENBERG: ijber eine Darstelhmg von Gruppenelementen in digitalen Rechenautomaten.<br />
Num. Math. 4 (1962), 151-153.<br />
2. W. L<strong>IN</strong>DENBERG: Die Struktur eines Obersetzungsprogrammes zur Multiplikation<br />
von Gruppenelementen in digitalen Rechenautomaten. Mitt. Rh- W. Inst. Znstr. Math.<br />
Bonn 2 (1963), l-38.<br />
3. J. NEUB~~SER: Bestimmung der Untergruppenverblnde endlicher p-Gruppen auf einer<br />
programmgesteuerten elektronischen Dualmaschine. Num. Math. 3 (1961), 271-278.<br />
4. See the papers by C. BROTT, R. BULOW, K. FERBER, V. FELSCH and J. NEUB~SER<br />
in these Proceedings.<br />
t at = ai= ai = aa = e, ala, = a,a,, ala3 = a3a2, ala, = afa,a,al, a,u, = a,a,a,,<br />
a2a4 = a,4a,, u3a4 = a4a$a2u1.
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