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
48 H. Jiirgensen Calculation with elements of a finite group Because of (Rl), (R2), and (R4) Uj/Uj_l is a finite cyclic group of order greater than 1; hence G is a finite soluble group. Now let G be a finite soluble group. If G is cyclic, it will be defined by the generator aI with glr = e and w1 = IGI. If G is not cyclic, there exists a finite chain of subgroups UO, UI, . . . , U, of G with: UO = gp(e); U,, = G; Uj_1 a Uj; Uj/ Ujel is cyclic and finite of order Uj : Uj-, =- 1 (1 =z j < n). For j = l(l)n aj is selected in such a way that gp( Uj-l, aj) = Uj; yj will be defined as Uj : Uj-1 ((R4) holds). Then the words gij can be found such that (Rl), (R2), and (R3) hold. For the proof that G is defined by an AGsystem chosen like this we define: Aword a”ma”~-1.. . aEvlc G(1 == vlen; eyi integer; i = l(l)n), for which the f%ov%g condi?ions hold, is called a normed word: Vi -c Vi+1; l=siim WI 0 =s EYi; l=si
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