COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

64 L. Gerhards and E. Altmann Automorphism group of finite solvable group 65
Fp of III,& From (1.11) we obtain:
Fk = fj Fkl= N(P1. . . . ‘P/+-lPk+I’ . . . -P,SG)nP,
is,
i*k
and according to (1.3) F = FIX . . . XFk is the Sylow system normalizer
related to the Sylow basis PI, . . . , Pr of G.
2. Determination of A(G) by composition of allowable automorphisms.
Let the composed mapping a = ~1. . . . � Z, with Zi E A(Pr) (i = 1, . . ., r) be
defined by
Then (cf. [l], ch. II, 0 1):
a(pl-... *Pr) = %Pl. . * * ‘%Pr Cpi E Pi>*
THEOREM 2.1. a 6 A(G) ifand onZy if
TCkOpikOTZ~‘=(Zipi)k A .
i, k=l, . . . . I
i+k
Proof Since PiPk = PkPi we obtain a E A(G) if and only if for i< k:
(1.7)
&kpi) = a(pkipi’pi kpk) = &k ipi)‘a(pi kpk) = (apk>*(api)
a.71
= (apk) l (api) * (api) k (apk),
hence : a(Pk iPi) = (apk) i(api) and 4Pi kPk) = (aPi) k (&Pk).
(2.1)
Because of the definition of a, these relations however are equivalent to (2.1).
From the point of view of computation it seems to be profitable to reduce
the number of the relations (2.1), for which we have to decide the equality.
The following theorem is fundamental for this reduction, and therefore for
the construction of A(G):
THEOREM 2.2. The sign of equality is validfor all relations (2.1) if and only
iffor all generators p? of a generating system {pr)} Of Pi the images of the
mappings nk 0 pi k 0 z;l applied on the elements pg’ and nkpp’ Of a generating
system {p$‘)} of Pk are the same as for the mappings (nipi) k (i, k = 1, . . . , r,
i + k).
Proof. It is sufficient to prove the relations :
nk 0 (p$“*p12)) k 0 7CF’ = (Zi (p$“.pj2’)) k (Pi% P12’ E {PI”‘}), (2.2)
nk 0 pi k 0 32:’ (P$‘*Pp’) = (7CiPi) k (Pp’*Pp’) (Pf’, Pf’ E {Pk’}). (2.3)
Then it is easy to give a complete proof of Theorem (2.2) by induction on
the powers of the generators pj’) and p$$ of {p?} and {pk)} respectively, and
by induction on the length of the words in those generators representing the
elements of Pi and Pk respectively. By the realization of this proof the same
calculation as below will show that (2.1) is valid on z&~)‘p~2’). We obtain
for k -K i:t
since pjl), pj2) E {pi*,)}
Zk 0 (pp)‘p$) k Onk1 (lzyf nk Opil) k oPy)k 07Z;l
proving (2.2), and because of
= StkOpy)k OZk-107CkOpf2)k OZ,l
= (jzip~‘)) k 0 (??Iip$2’) k
@/y) ((nip$‘)) (nip$2))) k
= (Zi(pj” ‘p’“‘)) k ,
Let~={aEA(G)/a=nl...:~~,niEA(Pi),(i=l, . . ..r))bethesetof
all automorphisms of G composed by automorphisms of the Sylow subgroups
Pi. Then according to the relation (cf. A, 9 1) g E F-z(g)Pi = Pi
(i = 1, . . ., r, -c(g) E I(G)) there corresponds to every coset decomposition
G=glF+... +g,F of G by Fa coset decomposition A(G) = z(gl)F+ . . . +
-t-z(g,)r of A(G) by r.
DEFINITION 2.1. An automorphism ni E A(Pr) (i = 1, . . . , r) is called
allowable if and only if there exists for every k + i an automorphism
ZkcA(Pk) such that a = Zl* . . . .Zi* . . . *Z,E A(G).
If ni E A(Pi) is allowable, then
@i(k) = F/k’ (2.4)
niNi’k’ = Ni’k’ (2.5)
% E N(&, k C SI ~6 1) (2.6)
n,(N*nP,) = N*nPt, N* char G. (2.7)
By every allowable automorphism 7ti an automorphism of the factor group
Pi/N!‘) E IIk, i will be induced, and fixing the index i and assuming Zi =
= eAVij we obtain by (2.1) for an allowable automorphism nk E A(Pk) (k 9 i)
the relations : zk Opi k 07Cz’ = pi k ; (nkpk) ipi = pk ‘pi, (2.8)
t In consequence of (1.9), similar results are valid for k =- i.