Intersecting subgroups of finite soluble groups (Q1108367)

[See also the preceding item.] For computational purposes a finite solvable group G is described by a power-commutator presentation (pcp) with generators passing through a chain \(G=N_ 1>...N_ r>N_{r+1}=1\) of normal subgroups of G with elementary abelian factors \(N_ i/N_{i+1}\). A subgroup or factor group modulo \(N_ i\) is described by an induced sequence of pcp-generators. There are collection processes which transform a given word into a normal form with respect to the given pcp. Algorithms are being developed that proceed by recursion from \(G/N_ i\) to \(G/N_{i+1}\); this is known to be efficient for finite p-groups. The two papers address the problems of finding a conjugating element x for conjugate subgroups and of constructing Hall subgroups, normalizers of subgroups and the intersection of subgroups H and K. Corresponding algorithms are proposed under additional hypotheses, namely when the conjugate subgroups are Hall \(\pi\)-subgroups, the normalizers are taken of Hall subgroups, and when every \(N_ i/N_{i+1}\) is covered by one of the intersected subgroups H and K or is central and avoided by H and K. A VAX implementation produced favourable running times on two groups \(G=((C_ 7 wr C_ 5) wr C_ 3) wr C_ 2\) and \(G=(S_ 4 wr S_ 4) wr S_ 4\) of orders 2 13 25 \(67^{30}\) and \(2^{63}3^{21}\) with short relations in the pcp. The intersection algorithm is generally applicable for p-groups and yields running times comparable to those for an existing p-group algorithm by Leedham-Green described in the SOGOS paper [\textit{R. Laue}, \textit{J. Neubüser} and the reviewer in Computational Group Theory, Proc. Symp., Durham 1982, 105-135 (1984; Zbl 0547.20012)]. Meanwhile the author and M. C. Slattery (preprint) have extended these algorithms to the general situation, where, however, they still rely on unfavourable orbit- stabilizer algorithms (SOGOS paper) in extreme cases. They are being implemented in the systems CAYLEY and SOGOS.