COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

80 W. Lindenberg and L. Gerhards Set of all subgroups of a finite group 81
position. The order I$!3 n B 1 is equal to or less than the greatest common
divisor of the orders of B and ‘B’. Thus assuming BA B’ =I= B resp. B’ the
order 1 {c(B, B’)}I of the compound {BV B’} will be bounded in the following
way by a number called MINORD:
/
[ l!l?l-123’j, if BAB’ = 0
2.max(19311, /%I), if 1% 1% Omod 1%‘1
I {‘B u B’} I + MINORD =
I
resp. I WI = 0 mod !Z 1
least common multiple of j % 1, I W I
I for all other cases (3.10.0)
If MINORD, computed by (3.10.0), is not yet a divisor of I&l, MINORD
is repeatedly increased by the least common multiple until /@&,I is divisible
byMINORD.IfthereexistsacharacteristicnumberB”(~“)EL1UL:(I~”j=
MINORD)-this may be tested by using 3.6.0-satisfying (B v B’) A B” =
BV B’ the composition may be omitted, for c(B, B’) would be equal to
B”; otherwise we are looking for an element B”(fI3”) E L1 IJ LT satisfying
(BVB’)AB” = BVB’ and I%“I=-MINORD. Similarly to 3.3.0, IW’I may
be used for bounding the generation {BV B’}. Summarizing, we note
3.10.0. By computing tee boundary MINORD of {!I3 U B’}, the process
of generating may be either omitted or bounded.
The method by which the remarks 3.3.0. to 3.10.0. are used for composition
is given in the flow-chart for computing c(B(@, B’(%?‘)) by B(8) and
B’(23’).
4. Conclusion. By means of the computed set L1 we are able to compute
the full lattice V(@) of all subgroups of 8. The method of computing V(G)
is above all a method of iterative reduction of the set of all characteristic
numbers by selecting-mainly by using (1.4)-those characteristic numbers
corresponding to maximal subgroups. Since this method has been described
in [2], it need not be further discussed here.
Finally the following table contains some examples of computed groups
together with some further information. The computing time includes not
only the time for computation of the lattice but also for computation of
conjugate subgroups, the normalizers and centralizers of the representatives
of these classes, and special characteristic subgroups, such as ascending
and descending central-chains, Fitting and Frattini groups and others.
--
- -
Order
60
120
168
192
192
216
900
.
L
L
Mode of generation I
Permutations of
degree 5
Permutations of
degree 5
Permutations of
degree 7
Permutations of
degree 8
Permutations of
degree 8
Permutations of
degree 9
2 abstract element
-
(
S
-
Number
of all
cyclic
subgroups
)f prime.
power
order
31
56
78
61
89
76
41
-
I
i
Number
of all
proper
subgroups
57
154
177
349
467
180
382
Number
, 3f classes
of conjugated
proper
SS
mubgroup:
w
7-
L
Computing time
7 - 14.4 set
17 1 min 12.6 set
13 3 min 15.3 set
56 4 min 40’0 set
76 20 min @2 set
18 5 min 28.5 set
110 18 min 27.0 set