COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.
COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA. COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.
134 R. Billow and J. Neubiiser Derivation of the crystal classes of Rq 135 matrix X = (xik) such that XGi=H$for i= 1, . . ..s. (4.1) This is a homogeneous system of linear equations for the xjk with coefficients in Z. A programme has been written which reduces (4.1) by integral row-operations to row-reduced echelon form. From this it determines the xjk as integral linear combinations of some x1, . . . , x, of them, chosen as parameters. Then it computes det X as a polynomial p(xl, . . . , x~) in these parameters. Three cases can occur: 1. If&l, . . .) x,) E 0 the mapping Gi -+ Hi is not induced by transformation with any matrix from CL,(Q) (where Q is the rational field). 2. If p(x1, . . . ) x,) $ 0 and the (integral) coefficients of all monomials in it have a greatest common divisor s 1, then no X with det X = f 1 can exist as the parameters can only be substituted by integers. In this case, however, Gi+Hi can be induced by transformation with some matrix X with det X + 0 and hence @ and @ are geometrically equivalent. 3. Ifp(x1, . . . . x,) 9 0 and the greatest common divisor of the coefficients of all monomials is 1, one has to try to find values of the parameters Xl, .a-, x, such that the matrix X obtained by substituting these values for the parameters has det X = 1. If one does find such values for the xl,. . . , x,, 8 and @ are arithmetically equivalent; if one does not find such values, only geometrical equivalence has been proved, but no conclusion about arithmetical equivalence has been reached. In all examples treated by us in which the third case occurred, it was always possible to find such values; so in fact the study of the greatest common divisor of the coefficients of the monomials ofp(xl, . . . , x,) was sufficient to prove nonequivalence. However, we owe to W. Gaschiitz an example of two groups (in GLz2 (Z)) for which the greatest common divisor of these coefficients is 1, but which are not arithmetically equivalent. 5. By the procedures described, we were able to classify the 1869 groups into both geometrical and arithmetical crystal classes. It turned out that the geometrical classes coincided with the similarity classes introduced here; this incidentally is also true for n = 1, 2, 3. Although it is unlikely that this is always the case, we do not know of an example of similar but not geometrically equivalent groups for any n * 5. 710 arithmetical classes were obtained, but as some hand-work was involved in the sorting, etc., this number has to be checked before it can be regarded as certain. Independent derivations of the arithmetical classes have been undertaken by H. Zassenhaus and D. Falk [lo] by slightly different algebraic methods and by H. Wondratschek by more geometrical means. It has been agreed that the results of all these calculations will be collated before they are published jointly. As a by-product of this investigation one gets a complete list of Bravais lattices in 4 dimensions which will correct an incomplete (and partially incorrect) list previously obtained [8] by heuristic considerations. The list of arithmetical classes will also be used to determine the list of all space-groups in 4 dimensions. A programme for this, following ideas of H. Zassenhaus [9], has already been written and used on partial lists of crystal classes by H. Brown. We would like to thank Professor H. Wondratschek, who first aroused our interest in the subject, for many valuable discussions and suggestions. REFERENCES 1. J. J. BURKHARDT: Die Bewegungsgruppen der Kristallographie, 2nd ed. (Birkhluser Verlag, Basel, Stuttgart, 1966). 2. E. C. DADE: The maximal finite groups of 4X 4 integral matrices. Illinois J. Math. 9 (1965), 99-122. 3. V. FELSCH and J. NEUBUSER: Ein Programm zur Berechnung des Untergruppenverbandes einer endlichen Gruppe. Mitt. Rhein-Westf. Inst. f. Znstr. Math. Bonn 2 (1963), 39-74. 4. C. HERMANN: Translationsgruppen in n Dimensionen. Zur Struktur und Materie der Festkiirper, 24-33 (Springer Verlag, Berlin, Giittingen, Heidelberg, 1951). 5. A. C. HURLEY: Finite rotation groups and crystal classes in four dimensions. Proc. Camb. Phil. Sot. 47 (1951), 650-661. 6. A. C. HTJRLEY, J. NEUB~~SER and H. WONDRATSCHEK: Crystal classes of four-dimensional space R4. Acta Cryst. 22 (1967), 605. 7. J. S. LOMONT: Applications of Finite Groups (Academic Press, New York, London, 1959). 8. A. I. MACKAY and G. S. PAWLEY: Bravais lattices in four-dimensional space. Acta Cryst. 16 (1963), 11-19. 9. H. ZASSENHAUS : Faber einen Algorithmus zur Bestimmung der Raumgruppen. Comm. Math. Helv. 21 (1948), 117-141. 10. H. ZASSENHAUS: On the classification of finite integral groups of degree 4. Mimeographed notes, Columbus, 1966. CPA 10
A search for simple groups of order less than one milliont MARSHALL HALL Jr. 1. Introduction. In 1900 L. E. Dickson [17] listed 53 known simple groups of composite order less than one million. Three more groups have been added to this list since that time. A group of order 29,120 was discovered by M. Suzuki [31] in 1960, the first of an infinite class, and one of order 175,560 was discovered by Z. Janko [25] in 1965 which appears to be isolated. Very recently Z. Janko announced that a simple group with certain properties would have order 604,800 and have a specific character table. The construction of a simple group of order 604,800 is given for the first time in this paper. The search for simple groups described here is not as yet complete. Approximately 100 further orders, all of the form 2”3*5”7’, remain to be examined. A number of people have helped me with this search. Dr. Leonard Baumert has helped with advice and computing. Dr. Leonard Scott sent me the proof of a formula on modular characters. But my main sources of help have come from Mr. Richard Lane and Professor Richard Brauer. For more than a year Mr. Richard Lane has carried out a large number of complicated computations on the IBM 7094 at the California Institute of Technology’s computing center. Professor Richard Brauer has been generous with help in references, correspondence, and conversations. The construction of the simple group of order 604,800 was carried out in August 1967 at the University of Warwick and at Cambridge University. Mr. Peter Swinnerton-Dyer was extremely helpful in writing on short notice a program for the Titan computer at Cambridge which finally confirmed the correctness of the construction. 2. Notation. List of known simple groups in the range. The notation for the classical simple groups used here will be essentially that used in Artin [l]. Here let GF(q) be the finite field with q elements where q = p’, p a prime. t This research was supported in part by NSF grant GP3909 and in part by ONR contract N00014-67-A-0094-0010. 10* 137
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- Page 79 and 80: 148 Marshall Hall Jr. otherwise no
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- 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
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A search for simple groups of order<br />
less than one milliont<br />
MARSHALL HALL Jr.<br />
1. Introduction. In 1900 L. E. Dickson [17] listed 53 known simple groups<br />
of composite order less than one million. Three more groups have been added<br />
to this list since that time. A group of order 29,120 was discovered by M.<br />
Suzuki [31] in 1960, the first of an infinite class, and one of order 175,560<br />
was discovered by Z. Janko [25] in 1965 which appears to be isolated. Very<br />
recently Z. Janko announced that a simple group with certain properties<br />
would have order 604,800 and have a specific character table. The construction<br />
of a simple group of order 604,800 is given for the first time in this<br />
paper.<br />
The search for simple groups described here is not as yet complete.<br />
Approximately 100 further orders, all of the form 2”3*5”7’, remain to be<br />
examined.<br />
A number of people have helped me with this search. Dr. Leonard<br />
Baumert has helped with advice and computing. Dr. Leonard Scott sent me<br />
the proof of a formula on modular characters. But my main sources of help<br />
have come from Mr. Richard Lane and Professor Richard Brauer. For<br />
more than a year Mr. Richard Lane has carried out a large number of<br />
complicated computations on the IBM 7094 at the California Institute of<br />
Technology’s computing center. Professor Richard Brauer has been generous<br />
with help in references, correspondence, and conversations.<br />
The construction of the simple group of order 604,800 was carried out in<br />
August 1967 at the University of Warwick and at Cambridge University.<br />
Mr. Peter Swinnerton-Dyer was extremely helpful in writing on short notice<br />
a program for the Titan computer at Cambridge which finally confirmed<br />
the correctness of the construction.<br />
2. Notation. List of known simple groups in the range. The notation for<br />
the classical simple groups used here will be essentially that used in Artin<br />
[l]. Here let GF(q) be the finite field with q elements where q = p’, p a<br />
prime.<br />
t This research was supported in part by NSF grant GP3909 and in part by ONR contract<br />
N00014-67-A-0094-0010.<br />
10* 137
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