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