COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

72 L. Gerhards and E. Altmann
ni E I? belongs to Ii(j+‘) if and only if en, ++ 7c, maps the class of the above parj+l
tition of { 1, . . . , IPi]} related to n Nik)into itself and an element z of every
kc1
k+i
other class into an element Z of the same class, and in addition for all
Pj$l E {Pio;)l} Of pj+l the appropriate relations (5.5) are satisfied.
(i) Determination of the group r. It is sufficient to describe the iterative
process for determining r in the first two steps:
The groups A’l’(Pi) (i = 1,2) (cf. flow chart) are decomposed by fi”’ :
A(l)(PJ = SZ~)I$~)+ . . . +nf’)Ii2)
A’l’(P2) = z~i”I(,~)+ . . . + n’2”“p.
(5.6)
Then zz$+).z~) E A(P1P2) (1 < ele sl; 1 ee2 es2) if and only if for all
ppj E (pp)} of Pk the mappings *) opp) o (n$@)-1 and (zp)p$@) i applied
to {p$“‘)} and {lips”} of Pi produce the same image:
&) oppl
I i o(ni”(‘)-1 = (?$dpk)i (ik=l 2.
, , 3 i+k). (5.7)
The test of (5.7) will be executed similarly as in (h). For the further investigation
there remain the following groups:
A@(JJ3 = 7@)1,(2)+ . . . +n$ks)@) (i = 1, 2).
Decomposing Ij2) by Ii”’ (i = 1, 2) we obtain a decomposition of A(‘)(Pi)
by Ii”‘. Now we decompose AC2)(P4 = A*(Ps) by Ii”‘. The test of the representatives
of these decompositions to be allowable automorphisms is now
the same for the pairs of groups AC2)(Pl), At2)(P3) and AC2’(P2), AC2)(P3) as
described for the groups All)(P1), AC1)(P2) in the first step. Continuing this
method for r steps we construct the group r.
SinceG=Pr... . *P,, each element of G can be represented as a word in
the special generators needed for the determination of A(Pi) (i = 1, . . . , r).
With respect to these generators it is easy to construct the inner automorphisms
z(g,) E I(G) generated by the representatives of the coset decomposition
of G by the system normalizer of G; and therefore the composition of
A(G) by J’ according to A(G) = z(gSr+ . . . +z(gr)r is obvious.
C. OPTIMIZATION AND EXTENSION OF THE
PROGRAM SYSTEM
6. Some aspects of optimization of the program system. Let G be a
solvable group and PI, . . . . P,aSylowbasisofG.IfG=QrX...XQ,is
a direct product of direct indecomposable factors Qi = Pi,l. . . . ‘Pi,,,
(i= 1, . . . . S; &ri ’ = r>, wh ere the Pi,k belong to the Sylow basis of G, it
can easily be proved that the direct product is uniquely determined.
Automorphism group of finite solvable group 73
The relations K[Pi~ * . . . *Pi,] = K[Pi,] V . . . V KIPiI] ({il, . . . , i,} S
c (1, . . ., r}) and the knowledge that both & = Pi,- . . . . P,, with
IKeI = jq
and the complemented group K@n = Pi,* . . . *Pi, with
are invariant in G and characterized in B,
0 3 (d), (6), enable us to develop a time-saving method for the determination
of the direct product G = QrX . . . XQ, of G. Since the automorphism
group A(G) is the composition of all automorphisms of Qi (i = 1, . . ., s),
we have only to apply the method described in 0 5 to the components
Qi (i = 1, . . ., s).
From A, 0 2 it seems to be profitable to alter the sequence of the groups Pi,l
(I= 1, . . . . ri) of the Sylow basis of Qi (i = 1, . . . , s), such that the index
[HP>: %(goJ] of the system normalizer of Qi in HyJ = Pi,j~. . . . *Pijl E Qi
(1=2,..., ri) is maximal, where the Pi,j~ (Y = 1, . . . ,1) are groups of the
Sylow basis of Qi. Then the order IJ’+)l of the group rH,& A(Qi) generated
by composition of allowable automorphisms Of Pi,j& (k = 1, . . . , r) will be
minimal, and therefore in general the iterative process for the determination
of I’,, will be optimized.
7. Some aspects for the extension of the program system to groups with
a normal chain of Hall groups. For the determination of A(G) of a finite
solvable group G we used only the following assumptions for the Sylow
basis of G: (a) G=P1-... *P,, PiPk = PkPj, (IPil, IPkI) = 1 (i, k = 1, . . . r;
i + k), (b) each Sylow system of G is conjugate in G. Therefore the
methods of B, § 5 for constructing A(G) can be extended to all groups G containing
a system of subgroups Hk (k = 1, . . . , s) such that G = HI- . . . -H,,
HiHk = HkHi, (IHi/, IHk/) = 1 (i, k = 1, . . ., s; i =I= k) and such that all
systems of that kind are conjugate in G.
In the case that G contains a normal chain of Hall groups :
there exists a system of subgroups Hi (i = 1, . . . , s) in G such that Hi 4 HiHk
for i-=k, Ni = H1- . . . *Hi, HiHk = HkHi and such that all systems of
that kind are conjugate in G (cf. [l], ch. II, § 2). The program developed in B,
8 5 therefore can be extended to all not necessarily solvable groups containing
a normal chain of Hall groups. If s = 2 we obtain the case that G is a group
extension of N by T with (INI, ITI) = 1:
G = NT, Na G, T z G/N, Nn T = {eo}, (IN\, ITI) = 1.
The automorphism group A(G) can be constructed in this case by composition
of allowable automorphisms of N and T.
6*