Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0707.20001
Glasby, S.P.; Slattery, Michael C.
Computing intersections and normalizers in soluble groups. (English)
[J] J. Symb. Comput. 9, No.5-6, 637-651 (1990). ISSN 0747-7171

Refinements of earlier algorithms by the first author [J. Symb. Comput. 5, 295-301, 285-294 (1988; Zbl 0654.20002; Zbl 0654.20001)] are presented that try to avoid orbit-stabilizer algorithms with computation of large orbits of cosets under right multiplications or of subgroups under conjugation. \par Soluble groups G are represented in a computer by a (consistent) power- commutator-presentation passing through a normal series $G=N\sb 0>N\sb 1>...>N\sb r=1$ with elementary abelian factors, subgroups have a canonical generating series, many computations proceed inductively from $G/N\sb i$ to $G/N\sb{i+1}$ [{\it R. Laue}, {\it J. Neubüser} and {\it U. Schoenwaelder}, Computational Group Theory, Durham, 1982, 105-135 (1984; Zbl 0547.20012)]. \par In order to compute the intersection of two subgroups H and K the authors now insert the subgroup $R\sb i/N\sb{i+1}=(H\cap N\sb i)$ $(K\cap N\sb i)$ $N\sb{i+1}/N\sb{i+1}$ and still use an orbit-stabilizer algorithm (for an affine group on a vector space, not for a general permutation group as before) to get from $HN\sb i\cap KN\sb i$ to $HR\sb i\cap KR\sb i$, but are then able to apply a generalization of the orbit-stabilizer- free covering algorithm [{\it S. P. Glasby}, loc.cit.] to calculate $HN\sb{i+1}\cap KN\sb{i+1}$ from $HR\sb i\cap KR\sb i$ (it does not cause trouble that $HR\sb i$ or $KR\sb i$ may not be a subgroup). \par Similarly for the computation of the normalizer of a subgroup H [{\it S. P. Glasby}, loc.cit.] new steps are inserted that reduce the computation of large orbits (on subgroups under conjugation) to a number of smaller orbits in connection with a variation of the orbit-stabilizer-free conjugation algorithm in coprime situations. [Further improvement is offerend in a recent paper by {\it F. Celler}, {\it J. Neubüser} and {\it C. R. B. Wright}, Some remarks on the computation of complements and normalizers in soluble groups, in: Topics in Computational Algebra, 57-76 (Kluwer 1990), reprinted from Acta Appl. Math. 21, No.1/2, 57-76 (1990)]. Implementations within the system CAYLEY (and GAP) demonstrate the effectiveness of these refined algorithms.
[U.Schoenwaelder]

MSC 2000:: *20-04 Machine computation, programs (group theory)
20D25 Special subgroups of finite groups
20D10 Solvable finite groups
68W30 Symbolic computation and algebraic computation
20B40 Computational methods (permutation groups)
20F05 Presentations of groups

Keywords: algorithms; computation of large orbits of cosets; Soluble groups; power- commutator-presentation; normal series; intersection of two subgroups; orbit-stabilizer algorithm; normalizer; CAYLEY

Citations: Zbl 0654.20002; Zbl 0654.20001; Zbl 0547.20012

Cited in: Zbl 0719.20010 Zbl 0708.20001

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]