COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

46 M. J. Dunwoody
is a chief series for G. Let hl, . . . , h,- 1 be a set of n- 1 generators for G.
Let(gl, . . . . g,J E Z, by the induction hypothesis on G/M1 there exists
a E A such that
kl, . . . , g&x = (m, m&l, m&t, . . . , m,-A-1)
where m, ml, m2, . . . , rnael c Ml. If m = e, then rnlhl, rnzhz, . . . , mn-lh,-l
generate G and by using a product of the ai:l ‘s and their inverses we obtain
a set of generators in which the first element belongs to MI and is not e.
Therefore it can be assumed that m =j= e.
Now, since MI is abelian, if
then
g = w(hl, hz, . . . . hn-l)EG
g-lmg = w(hl, . . . , h,- l)-lmw(hl, . . . , h,- 1)
= w(mlhl, , . . , m,- A- l)-lmw(mlhl, . . . , m,- Ih,- I).
It follows that there is an element a’ in A such that
(g1, . . . , g&' = W', mlh, . . . , mile A- 1).
Now using al:j+I or its inverse the (i+ 1)th term can be changed to timihi
or m-gmihi. However, since MI is minimal normal, each mi is a product of
conjugates of m or its inverse. Hence by repeating the above process enough
times it can be seen that there exists a” in A such that
(g19 . . ., g&d = Cm, hl, h2, . . ., h-1).
However hl, . . . , hnel generate G and so by using a product of the aizI’s
and their inverses we see that (gl, . . . , g,) belongs to the same transitivity
class under A as (e, hl, hz, . . . , h,), which proves the theorem.
To find a counter-example for the non-soluble case a computer might be
employed. If G is the alternating group on five symbols and Z is the set of
sets of three elements which generate G, then 2 has 120X 1668 elements
[l]. These are partitioned into 1668 transitivity classes under the action of
automorphisms of G, and A can be regarded as acting on these classes
rather than on the elements of 2. If A is not 19-ply transitive on these
classes, then the direct product of 19 copies of G, which can be generated
by two elements, would have a set of three elements which generate it but
which could not be reduced to two elements by Nielsen transformations.
REFERENCE
1. P. HALL: The Eulerian functions of a group. Quart. J. Maths. (Oxford Series) 7
(1936), 134-151.
Calculation with the elements of a finite group
given by generators and defining relations
H .J~~RGENSEN
1. Preliminary remarks. The system of group theoretical programmes
working at Kiel[4] consists of programmes which are independent of, and
others which depend on, the special way in which the elements of the group
to be calculated are represented. The latter are, roughly speaking, concerned
with reading the input data and printing, multiplication of, and inverting
elements.
I shall give an outline of some difficulties which arise with multiplication
and inverting programmes, when the elements are represented by words of
abstract generators, and of some ways to overcome or avoid them.
In 1961 Neubiiser [3] described a programme by which these problems
were, to a certain extent, solve8 for finite p-groups. In 1962 and 1963
Lindenberg published ideas [l] and a detailed description of a programme
[2] for solving them for finite soluble groups.
Thus in a certain sense no theoretical difficulties were left; but, as experience
proved, the practical problem of “minimizing” the time needed for
computing the product of two elements, when just the necessary input data
would be given, was not yet solved sufficiently. Hence some further refinements
had to be introduced.
2. Input data. Let G be a finite group and e the identity element of
G. An AG-system of G is a system of n generators a,, a,-,, . . . , al of G
and of n(n+ I)/2 words gij (i = l(l)i; j = I(l)n), for which the following
conditions hold :
gij E uj- 1 = de, al, a2, . . . , aj-1) G G; 1 =sisj=sn
(RI)
a? = gjj; 1 ej