COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

132 R. Biilow and J. Neubiiser Derivation of the crystal classes of R4 133
one representative from each of the classes of conjugate subgroups of the
nine groups and classify the set of subgroups of GL,(Z) thus obtained.
This has been done using a programme @ [3] developed at the "Rechenzentrum
der Universitat Kiel” for the investigation of given finite groups
by a computer.. This programme determines, for a finite subgroup @ of
GL,(Z) given by a set of generating matrices, among other things the
following :
1. a list of all elements of 8,
2. the classes of elements conjugate in @,
3. the lattice of subgroups of @, where for a representative U of each
class of subgroups conjugate in CV the following information is given:
(a) generating elements of U,
‘:
(b) all maximal subgroups of ll,
(c) for each class 0. of elements conjugate in U the number of elements
in 0. and their order, trace, and determinant.
The lattice of subgroups of the group Q4 exceeded the capacity of the
store of the machine at our disposal. For this group we first computed the
classes of conjugate elements with the programme di. From these it is
easily seen that the element -E (where E = unit element) is contained
in the intersection of the centre and the derived group and hence in the
Frattini subgroup of Q4. Hence -E is contained in all maximal subgroups
of Q4 and we found these by computing the lattice of subgroups of Q4/( - E)
which could be handled in our machine. There are three maximal subgroups
of order 576 normal in Q4, two classes of three conjugate maximal
subgroups each, of order 384 and one class of sixteen conjugate maximal
subgroups of order 72. One representative of each of these six classes of
maximal subgroups of Q4 and the other eight Dade groups were then investigated
with the programme @, giving a total of 1869 representatives
of classes of conjugate subgroups of these groups, which had to be classified
into crystal classes. This has been done in the following way:
Let us call two subgroups @, @ of G,%,(Z) similar, if there is a l-1
correspondence between the classes of conjugate elements in @ and those
in @, such that corresponding classes contain the same number of elements
and these have the same order, trace, and determinant. Obviously this
similarity is an equivalence relation, implied by both geometrical and
arithmetical equivalence.
The 1869 groups mentioned above were first sorted into similarity
classes using the information provided by the programme. 227 such
classes were thus obtained, consisting of 1 up to 43 groups. Geometrical
and arithmetical equivalence then had to be decided only within these
classes.
3. To some extent geometrical and arithmetical equivalence were treated
simultaneously. We therefore speak just of equivalence, if a distinction
is not relevant. By definition two subgroups @ and .$ of G&(Z) are equivalent
(geometrically or arithmetically), if there is an isomorphism v of
@ onto $$ and an integral matrix X (with det X + 0 or det X = + 1 resp.)
such that
XG = (Gv)X for all G E @. (3.1)
The work involved in checking this for the groups in each similarity class
has been substantially reduced by the following arguments:
1. An equivalence established between groups 8 and @ implies equivalence
between the corresponding subgroups of @ and @. We therefore
started deciding the equivalence of big groups and used the information
gained for smaller groups.
2. On the other hand, if a group @ contains a class of conjugate subgroups
Ui distinguished from all other subgroups of @ by their isomorphismtype
and the invariants of their elements, then a group @S* equivalent to C&j
must have again a unique class of subgroups UT with the same properties,
and the UI)s must be equivalent to the UF’s. This remark often allows US
to deduce nonequivalence of groups from nonequivalence of certain subgroups.
3. In order to prove nonequivalence of @ and .Y$ it suffices to show that
for some set of elements G1, . . ., Gk E C!i there is no set of elements Hr,
. . ., Hk E ,$ such that XG( = HiX holds for some X (with det X + 0 or
det X = + 1 resp.). For this purpose preferably Gr, . . . , Gk were chosen
as a class CS of conjugate elements, which is distinguished from all other
classes of @ by the invariants determined by the programme @ (number of
elements in B, order, trace, determinant). A group @* similar to csj then
contains just one class E* with the same properties. If %* is small enough,
a programme has been used to try all j&J*\ ! permutations of the elements
of B* as possible images of the elements of C$ in a fixed order. A similar
procedure was sometimes applied for two classes with either different or
equal properties.
4. In order to prove equivalence, one has to show that for some system
of generators Gr, . . . , G, of 8 there is a system HI, . . . , H, of generators
of@such thatXG, = H$(i = 1, . . ., s) for some matrix X. This was done
in the following way: A subgroup lI of order as low as possible and an
element B of order as high as possible were chosen such that U and B
together generate @ and that both the classes of conjugates of lI and of
B are unique in @ with respect to isomorphism-type and invariants. If 8*
is equivalent to @ there is a subgroup ll* and an element B* of a)* having
the same properties as U and B in 8. It then suffices to fix lI* as image of
U and to try all conjugates of B* as images of B.
4. All the choices described so far lead to some sets of elements Gi, . . . , G,
of @andHI, . . . . H, of @ for which we have to decide if there is an integral