COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

76 W. Lindenberg and L. Gerhards Set of all subgroups of a finite group 77
(‘% !8 C @), we define
B(%)VB(~) = ‘E’(ajVbj)*2j-‘,
j=l
B(%)AB(%J) =‘f’(aj,bj)*2J-l,
i=l
(1.1)
where aV b resp. ai\ b are the usual boolean operators. In particular we
have [4]:
B(%n%) = B(B) A B(B) (1-2)
B(W u ‘B}) == NW v N’B)+ (1.3)
9lE5.3 * W20 A B(B) = N90. (1.4)
2. The generation of the partial set of subgroups. In 0 1, the set of the wanted
subgroups of @ was represented by a set of certain numbers of l&i digits.
Let S be the normal sequence of the first 2!=nl binary numbers. To each
element of S belongs-as discussed above-a well-defined partial set G
of 2,. By (5 there is also defined a subgroup 9X = (6,) of @i, which is generated
by all elements of e,; it is GC 9X n 2,. Thus theoretically we can obtain
the wanted set of all subgroups of 8 by taking all elements of S, constructing
the corresponding subsets of Z,,, generating the groups defined by these
subsets and finally storing their characteristic numbers, if they have not yet
been determined. But a method of this kind cannot be used because of the
large number 21znl of operations to be done.
Therefore in [2] a method has been described for systematically reducing
the above set S of binary numbers used for the determination of the desired
set of all subgroups of @5. Beyond this, however, it is practicable to divide
the listed elements of 2, into s sections Ci (i = 1, . . . , s) of length 2 + lZ,l
and one further section C,,, of length r (with lZ,J = s*Z+r(r-= r);
c s+l = 0, if r = 0) wr‘th ou tpermuting
the sequence of elements of Z,, i.e.
z, = [Zl, * . -3 ZlZ,ll
= fz19 * * * 3 212 . - * 9 zif+12 * * * 3 Z(i+l)f, - - * 7 Z(s-1)1+1, * * + 3
Zsl, Zsl+l, * * -, Zsr+#
If now the method of [2] is used for each of these sections Ci (i = 1, . . . , sf 1)
we obtain for each Ci a set Gi of subgroups of 8. Generally any two of these
sets will not necessarily be disjoint, i.e. the characteristic number may occur
more than once, but by eliminating those multiple elements, we obtain the
disjoint sets Gj. Naturally in general Ui Gi is not yet the wanted set of all
subgroups of (8. In 0 3 we give a detailed description of the method of combining
the characteristic numbers.
t The equality does not hold, because generally 2X U bc{% U S}.
$ Empirically it seems useful to choose 6 G I =S 9.
3. The method of constructing the set of all subgroups of a. First we give
some definitions and notations :
DEFINITION 1. The determination of the characteristic numbers of the
group Q = {‘$.I, !-B} by the characteristic numbers B(‘9.l) and B(8) is called
composition; the characteristic number of 6 is denoted by c(B(%), B(8)),
the order of 6 by lc(B(%), B(B))].+
DEFINITION 2. Let Sbe a set of characteristic numbers; Sis said to be closed
(relative to composition) if c(Br, Bs) E S for arbitrary elements Bl, BzE S.
DEFINITION 3. Let S be a (not necessarily closed) set of characteristic numbers.
The closed set C(S) 2 S is called the closure of S.
We shall denote by
Ej the set of characteristic numbers of all subgroups of Gi (i = 1, . . . , S+ 1)
(see Q 3,
\,?&I the number of elements of Ej,
&,a,...,i =C(ElU . . . U&J-(EIU . . - UG)
El 1 2 ,...,I, ’ = C(EllJ . . . IJEi)
IJL2,. . . . i] is the number of elements of El,2,. . ., i
K = {c(B,B’) 4 Et U El z
, ,...I
Clearly we have:
i-11 BE& B’CEl,2,. ..,i-1 arbitrary
(i = 2, . . ., s+l)}.
c(c(B1, Bz), c(B;, B;)) = c(c(BlVB;), c(BzV@), (3.1)
Ei = C(Er’) (i= 1, . . . . s + l), but in a way we may also assume Ei = C(Ei)*
Now we prove
(3.2)
3.1. If BA E Ki (A = 1,2; 2 s i 4 sf l), then there always exist BTE Ei and
B,*EEl,2,...,i-l such that c(B1, Bz) = c(B,*, B$).
Proof. BnE Kt implies by definition Bn = c(BA, B;‘), B;E Er, B;’ E El, 2, . . . , t-1
(A= 1,2). Hence using (3.1): c(Bl, Bz)=c(c(B;, Bi’), c(Bi, B;‘))= c(c(B;V B;),
c(Bi’V B;‘)). According to (3.2) we obtain: c(BiV B.$ = Bf E Ei, according
to the definition of E,,2,. . . , f-1: c(B;’ V Bi’) = B.f E El.2,. . . , i-1. Thus
c(B1, B2) = c(Bf, B,*), as required.
Using 3.1 we immediately obtain:
3.2. KiiJEi!JEl 2
, ,..., i-1 is the closure of El IJ . . . U Ei (i = 2, . . . , s + 1).
Now we are able to describe the method of constructing the set of all subgroups
of 8. From (3.2) we have C(El) = El. Suppose C(ElU . . . UEr) =
K1, 2,. . . , i U E1 U . . . U Et = E1, 2,. . . , i has already been determined.
f For abbreviation we shall often write B1, B,, . . . , B’, B”, , . ., etc., instead of
W%), B(b), . . . .