COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

74 L. Gerhards and E. Altmann
REFERENCES
1. E. ALTMANN : Anwendungen der Theorie der Faktorisierungen, Forschungsbericht
des Landes Nordrhein Westfalen, Heft 1902. Zugleich Schriften des Rhein - Westf.
Inst. fiir Znstr. Math., Bonn, Ser. A, No. 20.
2. V. FELSCH and J. NEOBUSER: Uber ein Programm zur Berechnung der Automorphismengruppe
einer endlichen Gruppe. Num. Math 11 (1968), 277-292.
3. L. GERHARDS and W. LINDENBERG: Ein Verfahren zur Berechnung des vollstlndigen
Untergruppenverbandes endlicher Gruppen auf Dualmaschienen. Num. Math. 7
(1965), l-10.
4. P. HALL: On the Sylow systems of a soluble group. Proc. London Math. Sot. (2) 43
(1937), 316-323.
5. P. HALL: On the Sylow normalizers of a soluble group. Proc. London Math. Sot.
(2) 43 (1937), 507-528.
6. W. LINDENBERG and L. GERHARDS: Combinatorial construction by computer of the
set of all subgroups of a finite group by composition of partial sets of its subgroups.
These Proceedings, pp. 75-82.
7. W. LINDENBERG: Uber eine Darstellung von Gruppenelementen in digitalen Rechenautomaten.
Num. Math. 4 (1962), 151-153.
8. L. R~DEI: Die Anwendung des schiefen Produktes in der Gruppentheorie. JournaC
reine angew. Math. 188 (1950), 201-228.
9. J. SZEP: Uber die als Produkte zweier Untergruppen darstellbaren endlichen Gruppen.
Corn. Math. Helv. 22 (1949), 31-33.
Combinatorial construction by computer of
the set of all subgroups of a finite group by
composition of partial sets of its subgroups
W. LINDENBERG and L. GERHARDS
THE following paper contains a description of the main parts of a program
for computational determination of the lattice of all subgroups of a finite
group a. The program has been developed by the authors and has been
realized for the computer IBM 7090, It is of combinatorial type and does not
require further assumptions for @.
1. Preliminaries. Let @ = {X1, . . ., xn} be a finite group generated by an
ordered system (Xl, . . ., x,) of independent? elements (e.g. permutations,
matrices or abstract elements together with the connecting relations of
multiplication [l], [3]). Generating @ successively by x1, x2, . . . , x,, we obtain
achain(e)c@lc.. . C @), = @ Of subgroups Of @(pi = {Xl, . . . , Xi},
i= 1, . ..) n). The order of @i will be denoted by I@il. If !J,Ri is the set
of all cyclic subgroups of pi of prime-power order, Zi shall be some set of
generators of all elements of pi such that Zin Zk = Zi for all k == i
(i, k = l,, . . . , n). Let lZil be the number of elements of Zi. We shall list
all the elements of Zi (i = 1, . . . , n) in such a way that the sequence of the
elements in the list of Zi is the same as in the list of all elements of 2,
(kz= i), i.e. the part of the first ]Zil elements of the list of all elements of Zk
is identical with the list of Zi.
For characterizing any subgroups % of @i [4] we note that Z&f = ‘%n Zi
is uniquely determined by a. Now, if the @h element of Zi (14 9 == IZiI) is
associated with the binary number 2Qm1, and any subset G of Zi with the
sum B(G) of those binary numbers associated with all the elements of G,
we have that $!l corresponds uniquely with B(Z@). For standardization,
however, we complete this number by apposition of lZ,l- IZil zeros to the
higher digits of a number B(a) of IZ, 1 digits. B(a) shall be used for represent-
IZSl IZRl
ing the subgroup $!l c 8.x If B(a) = 1 aj.2’-’ and B(B) = ,g1bj*2’-’
j=l
t This is no loss of generality.
$ For the following discussions it will be useful to remember that, for any subgroup
xc&$ (NQ@$+ ,), B(3) may contain ones only in its /Z,( lower digits.
75