COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

68 L. Gerhards and E. Altmann Automorphism group of jnite solvable group 69
A system xl, . . . , x,, is called a special generating system of G if and only
if for every g E G we obtain a representation g = w(Xi) = ~1. . . . *XT: with
o=s&k=syk (k = 1, . . .) n), where yk iS the least power of + such that
qt = qb”. . . . xpp.
If G is solvable, we can obtain a special generating system x1, . . . , x, of G
using a subinvariant series
G = G, D Gnel D . . . D GI D Go = {ec} (4.4)
such that {XI, . . . , xi} = Gi and xi E N({x~, . . . , Xi-r} s G). The defining
relations of this system of generators are:
xri I = XY.“... . “yq’ (i= 1, . . ..n)
(L.0 0.0
.YkXi = x; - . . . *.&l Xk (1 d i == k == n).
(4.5)
Sometimes it is necessary to compute the automorphism 0: E A(G)--stored
by the images axl, . . . , ax, of the generators x1, . . . , x,, of G-as images
of another system yr, . , ., yr of generators of G. Assuming the relations
Yj = W,,(Xi) we obtain ayj = w,,(axi).
5. The computational method for the composition of A(G).
(a)+ Determination of a complete Sylow basis P1, . . . , P, of G. For all
KjEsQ(i= 1,. . . , r) (cf. A, 6 1, (a)) we obtain jKil = h @. In the OL-list we
j=l
j+i
search for the order iK,/ (cf. B, 0 3, (d), (,9)), and we find in the corresponding
place in the CL-list (cf. B, Q 3 (d), (a)) the characteristic number K[Ki]++Ki
(i= 1, . . . . r).Since Pi = 6 Kj (i = 1, . . . . r)wegetby(3.2):
j=l
i+i
Pi c* K[Pi] = i\ K[Kj] (i = 1, . . ., r). (5.1)
j=l
i*i
(b) Determination of the system normalizer P. Using (1.2) and (3.2) we
obtain K[FJ = A K[N(Ki E G)]. K[N(Ki !Z G)] (i = 1, . . . , r) is known
i=l
from the NL-list (cf. B, 0 3 (d), (y)).
(c) The coset decomposition of G by UC G and the determination of a systen2
of representatives % = {rl, . . ., rt} of the decomposition are basic
programs of the program system mentioned in B, 6 3 (a), (E).
(d) Some remarks about determination of V(Pi) and A(PJ (i = 1, . . . , r).
There is GI UC V(P,)-K[U] A K[Pi] = K[U]. For the determination of the
t The notations (a), (b), . . . , of Q 5 correspond to the marks on the flow chart (cf.
p. 71).
,
classes of conjugate subgroups OfPiwe mention : If U a G, UC Pi(Cf. B,$3 (d),
(6)), then U a Pi. If Pi contains a complete class S of subgroups of G, which
are conjugate in G, and if ISI = [Pi:N(REPi)] for some representative R of
the class S, then all U E S are also conjugate in Pia If IS/ + [Pi :N(Rc Pi)] = 1,
then R 4 Pi. If ISI =# [Pi : N(RE Pi)] + 1 we decompose Pi in cosets by
N(R E Pi) and transform R by the elements of a system of representatives of
this coset decomposition, using the relation
K[N( R C Pi)] = K[N(R C G) n Pi] = K[N(R E G)] A K[Pi] .
The determination of A(Pi) can be executed by the method developed in
the program system [2].
(e) Determination of the permutation groups fli, k (i, k = 1, . . . , r; i + k).
TheelementsOfPi(i= 1, . . . . r) are numbered in the same sequence as
they are generated by the generating program of Pi. Generating the subgroups
Gi,k = PiPk = PkPi Of G (i, k = 1, . . . , r; i< k) on the one hand
as a product of Pi, Pk and on the other hand as a product of Pk, Pi we
obtain by comparing the products:
p!“)pt) = p~)p~) = p~‘k&)*pf)ip~) (S = I
1, . . ., IPkl; 1