COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

70 L. Gerhards and E. Altmann
(a) Let F/(1 = 1, . ..) Ai) be the different fix groups of I&, k (i, k =
1 . .,r,i + k)andletN@)nP,(Y = 1 vi) be the different characteristic
s;bgroups of Pi induced by the N@)ciai*i’and also different from all characteristic
subgroups of Pi used by the determination of A(PJ. Each group
Fj’), NO n Pi determines a partitiont 8.;a) = [Fp, pi-@)], @ = [NWnpi,
Pi-(iV(“‘n Pi)] of Pi respectively. TO the partition
there corresponds a well-defined partition of the set { 1, . . . , 1 Pi I} :
(il, . . .,“ieJ, &fl, . . -,“iq+eJ, *. .Y t.ie,+ *..+.$-1+1,-. .,.iq+...+e*)
( ilez= lpi/) (5.3)
Now we calculate the permutation Q~< of the set { 1, . . . ,I Pi I} related to the
automorphism zi C A(Pi) given by the images nip?) of a generating system
{pJ’)} of Pi (cf. B, 0 4). Then 3ti E A*(PJ if and only if en, maps the classes of
the partition (5.3) complexwise into themselves.
(4 If fyi = wy{(pp),fy& = w$~gJp), . . . , fyi. = wyjJpp>
fy”$ = qy(p!P)),),~~ , t = qQp,!P)) , t 9..*, f;ui. ,t = +~~i(pp9;
is a generating system of GA), N(“)n Pi respectively - represented as words
of the elements ~$9’ of a generating system {#} of Pi - then ni E A*(Pi) if
and only if both
?Zjf$f) = Wj$ (nipi”) E F? for A = 1 ,***, 3.. ‘I, j= 1, . ..yhi
and niJ’I,“J = Wj,‘: (PZ&‘)) EN(“) n Pi for v = 1, . a ey Vi; j = 1, . . ., hi.
(h) Determination of I!” I (i = 1 , ..*, r; j = 2, . . .) r; i=i=k)(cf. flow
chart).
(a) Determination of I/j) (1 6 i =S r ; 2 =~j =S r). Let 8j“) (k = 1, . . . , j; i + k)
be the coset decomposition of Pi by Ni@), then, similarly to (g) (5.3), the
greatest common refinement /( 8;“) = pj”h N!k)+ . . . +p$“)b N{‘) correk=l
k=l
k+i
;;i’
sponds to a well-defined partition of the set:;: . . . , jPij}, which can easily be
determined. Those Xi E A*(Pi), the corresponding permutations eni of which
map the class of the partition of {I, . . . , 1 PiI } related to h N’k) into itself
k=l
k*i
t A non-empty system 8 of non-empty subsets Ns M of a set M+ 0 is called a partition
of M if and only if each element of M is contained in only one set N E 8. The subsets
NE M are called the classes of the partition 8, The greatest common refinement of the
partitions a1, 82 of M formed by all non-empty intersections of the classes of B1, se
is denoted by & x &.
Automorphism group of finite solvable group 71
and a representative z of every other class into an element Z of the same
class, belong to the group 1:” if and only if for all pp’ E {pp’} of Pk (cf.
Theorem 2.2) :
njOpkiOnlT1= pkt A .
k=l,.. ..j
k+i
(5.4)
The test of e quality for the relations (5.4) however is equivalent to comparing
multiplication of permutations in an abbreviated form (cf. Theorem
2.2):
C?n$k ’
.
en
-l=pki A , (5.5)
k=l,...,j
k+i
for the test of (5.4) is restricted on the generating systems {p/“)} and {nip{‘)}
of Pj.
(/l) Determination of Ii(j+‘) using&@. By use of the coset decomposition $+”
of Pi by Nyfl), the partition of { 1, . . . , IPil} related to the partition i 8$k),
k=l
k+i
which has been needed by the construction of Z?, can be easily refined.