Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0654.20002
Glasby, S.P.
Intersecting subgroups of finite soluble groups. (English)
[J] J. Symb. Comput. 5, No.3, 295-301 (1988). ISSN 0747-7171

[See also the preceding item.] \par For computational purposes a finite solvable group G is described by a power-commutator presentation (pcp) with generators passing through a chain $G=N\sb 1>...N\sb r>N\sb{r+1}=1$ of normal subgroups of G with elementary abelian factors $N\sb i/N\sb{i+1}$. A subgroup or factor group modulo $N\sb i$ 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 $G/N\sb i$ to $G/N\sb{i+1}$; 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 $N\sb i/N\sb{i+1}$ 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 $G=((C\sb 7 wr C\sb 5) wr C\sb 3) wr C\sb 2$ and $G=(S\sb 4 wr S\sb 4) wr S\sb 4$ of orders 2 13 25 $67\sp{30}$ and $2\sp{63}3\sp{21}$ 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 [{\it R. Laue}, {\it 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.
[U.Schoenwaelder]

MSC 2000:: *20-04 Machine computation, programs (group theory)
20D10 Solvable finite groups
20F05 Presentations of groups
68W30 Symbolic computation and algebraic computation
20D30 Series and lattices of subgroups of finite groups
20D15 Nilpotent finite groups

Keywords: finite solvable group; power-commutator presentation; generators; pcp- generators; collection processes; finite p-groups; normalizers of subgroups; intersection of subgroups; algorithms; Hall $\pi $ -subgroups; CAYLEY; SOGOS

Citations: Zbl 0654.20001; Zbl 0547.20012

Cited in: Zbl 0708.20001 Zbl 0707.20001 Zbl 0654.20001

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.
	Elementary number theory. Primes, congruences, and secrets.

Simple Search

Zentralblatt MATH Berlin [Germany]