COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

84 K. Ferber and H. Jiirgensen
The input for the programme rl is a paper tape with the following information
about the lattice L(G) of subgroups of G:
(1) the number k of classes of subgroups conjugate in G,
(2) the number n of subgroups of G,
(3) for each class Ki of conjugate subgroups Ui,, . . . , Ui?,, of G:
(a) the order of these subgroups,
(b) the list-numbers ii, . . ., i,,,
(c) for each Via E Ki the list-numbers of its maximal subgroups in the
numbering mentioned above.
A paper tape with this information is provided, for example [l], by a programme
CD implemented on an Electrologica Xl, which determines the lattice
of subgroups of a group G from generators of G. The programme rl which
has been implemented on a Zuse 222 reads this tape and punches a data
tape for the plotter Zuse 264. For this the programme rl needs some additional
information about the size and shape of the diagram wanted.
The following data for the drawing may be prescribed :
(1) the radius r of the circles representing subgroups (or half the side of
the squares),
(2) a (common) ordinate for the centres of circles representing a class Ki
of conjugate subgroups,
(3) an abscissa of these centres for each subgroup.
If these data are not prescribed, the programme A puts r = 3 mm and
tries to find suitable ordinates and abscissas. This is done in the following
way.
An ordinate is calculated as a function which depends linearly on the
radius r and logarithmically on the order of the subgroups in Ki. The abscissas
are calculated by the programme only under special conditions : G must
be a p-group or a group of order pmq, where p and q are primes and p-= q;
moreover, when U, V are subgroups of G, U maximal in V, 1 UI = p’q, 1 VI =
p’q, it is not allowed that there exists a subgroup W of G with 1 W 1 = ptq
and r-= t-= s. Geometrically this means that the diagram of the lattice of G
must consist of at most two “branches”.
For groups satisfying these requirements the set of all subgroups of order
pj, 0 ~js m, is called the first branch B1, the set of all subgroups of order
pjq is called the second branch B2 of L(G); the set of all subgroups whose
order is the product of i primes is called the ith layer L, of L(G). Li n Bj is
called the row R{.
In order to calculate the abscissas for B1, the row R& containing the
greatest number of subgroups is determined, and the abscissas for these
subgroups are defined from left to right according to the sequence in which
they occur on the data tape. All other rows are arranged in such a way that
their geometrical centre has the same abscissa as the one of Rf;.
,
A programme for the drawing of lattices
FIG. 2
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The abscissas for the subgroups in B2 are determined in two main steps.
First, consecutive rows of the first branch are considered. Let U be the
first subgroup in the row Rt-_,, V be the last subgroup in Rt, W be the first
subgroup in Rf . Then the abscissa x, for W is determined in such a way
that Y is left of the line connecting U and W and has a certain distance
from it (see Fig. 2). From x, the abscissa xCi of the centre of RF is found.
This calculation is done for all i, 1 e is m, and the maximum x of the X,i
found. Then all rows Ry are moved to the right until the abscissas of their
centres coincide with x.
Second, a similar procedure is performed for consecutive rows of the
second branch. Let U be the last subgroup of Ri-_,, V be the first subgroup
of Rf-, and W be the last subgroup of RF (see Fig. 3). It is tested if Vis to
the right of the line connecting U and Wand has a proper distance from it.
If not, the second branch is again moved to the right until this is the case.
This procedure is performed for all i, 1 -= i < m + 1.
From the maximal abscissas and ordinates the size of the diagram to be
drawn is obtained. If this is too large, the radius r chosen for the circles is
U e R;-,
FIG. 3
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