COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

62 L. Gerhards and E. Altmann Automorphism group of finite solvable group 63
every qi a qi-Sylow-complement Ki (i = 1, . . . , r) of order 1 Kil = fi q?, and
every complete system R = KI, . . . , K, of qi-Sylowcomplements generates
a complete Sylow system 6 consisting of 2’ subgroups K, = n Ki,
iC e
K$ = G, defined by all subsets q of the set of integers (1, . . . , r }. If q’
denotes the complemented set of ,q relative (1, . . . , r}, then lKel = n
jC e’ @ and
for Q, d C (1, . . ., r} we obtain the relations
j=l
j#i
(a) &“a = KenKo (1.1)
(PI Keno = K,K, = K,K,.
Every two Sylow systems %, e* of G defined by 9, R* are conjugate in G,
and every Sylow system 6 of G contains a complete Sylow basis, i.e. a system
PI, . . . , P, of Sylow subgroups of G such that G = PI . . . . . P,,
PiPk=PkPi(i,k= 1, . . . . r; i =/= k). Additionally we obtain K, = n Pi
te E (1, . . ., r}) for all K, E G.
The system normalizer ‘%(G) defined by Z(G) = {x E G/x-lK& = Kp,
for all K, E e} can be represented as the intersection of the normalizers
N(Ki C G) or N(Pi E G):
iEe
‘S(G) = h N(KiS G) = h N(Pi E G). (l-2)
i=l i=l
m(@is the direct product of its Sylow subgroups Pin N(KiC_ G) (i= 1, . . . , r):
%(G) = PI~N(KIEG) x . . . xP,nN(K,GG). (1.3)
An automorphism a E A(G) of G maps the Sylow basis PI, . . . , P, of G on a
conjugate one P1*, . . . , Pf, that means there exists an element g E G such that
aPi = Pjl: = t(g)Pi (i = 1, . . ., r; z(g)EZ(G)).t (1.4)
The automorphism /I = r(g-l) o a E A(G) maps Pi onto Pi (i = 1, . . . , r),
and the restriction of /3 on Pi yields an automorphism rCi E A(Pi) of Pi.
(b) General products ([S], [9], [l], ch. I). A group G is called a general
product of the given abstract groups Hi (i = 1,2) (or factored by Hi) if and
only if Gcontains two subgroups Hi* such that Hi* z Hi and G = H,*H,* =
= H,*H,*, Hfn Hz = {eo}.
Let G be factored by Hi (i = 1,2), then to each hi E Hi there corresponds
a mapping IZi k from Hi into Hi defined by:
h12 hz = H,hzhlnH, for all hz E Hz, (1.5)
ha 1 hl = hzhlHzn HI for all hl E HI. (1.6)
t We denote by z(g) the automorphism of the inner automorphism group Z(G) of G
induced by transformation by g E G.
The mappings hi’ (i, k = 1, 2; i + k) together with the defining relations
of Hi (i = 1,2) determine the structure of G; for multiplying the relation
(h;lh2H1 n Hz) * hl 2 hz = ec with hzhl from the left we obtain the following
law for changing the components of an element of the general product G:
hzhl = ha 1 hl-hlz hz. (1.7)
By (1.7) multiplication in G is completely determined.
Conversely, if HI, HZ are given groups and if according to (1.5), (1.6)
each hi E Hi (i = 1, 2) is associated with a mapping hi k (i, k = 1, 2; i =f= k)
from Hk into Hk, then the set G = {(hl, hz)/hi E Hi, (i = 1, 2)) of all
pairs (hl, h2) of elements hi E Hf with the multiplication law
forms a group if and only if the following relations are valid :
(a) elzhz = hz e2 1 hl = hl
(B) hl2e2 = e2 hz 1 el = el
(y) (hl-h;)2hz = h;2@12hz) (h2.h;) I hl = hz I (h; 1 hl)
(6) hlz(h2.h;) = (h$h$hz-hlzh; hzl(h,-h;) = h21hl.(h12h2) 1 h;.
(1.8)
(1.9)
The correspondence hl- (hl, e2), ha *-* (el, h2) determines the isomorphism
HI= H;” = {(hl, ez)/hl E H,}, H2r Hz* = {(el, h2)/h2 E HZ} respectively.
Because of G = H,*H,*, H: n H,* = {(el, e2)} = {eo}, hj’c k hz =
= (hikhk)*,G’ IS a g eneral product and conversely every general product can
be represented in this way.
From (1.9 a, 7) it follows that the mappings hik form a permutation subgroup
ni,kc SbH1.j of the symmetric group SIHkl of degree (Hkl. The maximal
invariant subgroup Ni = {hit Hi/hikhk = hk, for all hkc Hk} of G contained
in Hi determines the homomorphism Zi, k : Hi +IIi, k from Hi onto
II,, k. Hence :
Hi/N, S ni, k, (1.10)
and the mappings (hini) k with Izi E Ni define the same permutation on Hk.
Another important subgroup of the general product G is the “fix group”
Pi of UI,, i defined by Fi = {hi E Hi/hki hi = hi, for all hk E Hk}, containing
all elements hi of the component Hi, which are invariant against all map-
pings hk i related to all elements hk E Hk. For Fi we obtain:
Fi = N(Hk C G)n Hi; N(Hk C G) = FiHk = HkFi. (1.11)
If G is a solvable group with Sylow basis PI, . . ., P,, the theory of general
products is applicable to the subgroups Gk,i = PkPi = PiPk of G.
Important for the construction of the automorphism group A(G) are the
maximal invariant subgroups Nf) of Gk, i contained in Pk and the fix groups