COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

60 V. Fe&h and J. Neubiiser
2. The coarsest equivalence relation, - say, on the set of all subgroups
of G, with the following properties is constructed:
(1) Each group of the lower and upper central series and of the commutator
series of G forms a complete --class.
(2) U-Vimplies: jU/ = IV/; No(u)-No(V); Co(U)-Cc(V); Uand V
are both or are both not cyclic, abelian, nilpotent, supersoluble, soluble,
perfect, normal in G, subnormal in G, or selfnormalizing ; for each - -class
9, U and V contain, are contained by, and normalize the same number of
subgroups belonging to R! respectively.
3. For each element g E G the set H(g) of all h E G with (h)-(g) is determined.
H(g) contains the set J(g) of all images of g under A(G) and can be
shown to be a “good approximation” of J(g).
4. Generators bl, . . . , b, of G with minimal d = fI IH( are selected.
Then a list L of d bits is set up in 1 - 1 correspond&Ae to the d different
systems b;, . . . , bj with b(i E H(bj). Systems b;, . . ., bi not generating G are
marked in L.
5. Generators yl, . . ., p)m of the subgroup I(G)