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Computing projectors, injectors, residuals and radicals of finite soluble groups. (English) Zbl 0990.20008

The author presents algorithms for computing subgroups of a group related to Schunck and Fitting classes. They can be considered a generalization of methods of B. Eick and C. R. B. Wright for formations [J. Symb. Comput. 33, No. 2, 129-143 (2002; Zbl 0995.20004)]. As an application, the author obtains a method for computing normal subgroups for a solvable group, which improves on the algorithm of the reviewer [Proceedings of the 1998 international symposium on symbolic and algebraic computation, 194-198 (1998; Zbl 0943.20005)].

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
68W30 Symbolic computation and algebraic computation
20-04 Software, source code, etc. for problems pertaining to group theory

Software:

CRISP; GAP
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References:

[1] Butler, G., Fundamental algorithms for permutation groups, (1991), Springer Berlin · Zbl 0785.20001
[2] Celler, F.; Neubüser, J.; Wright, C.R.B., Some remarks on the computation of complements and normalizers in soluble groups, Acta appl. math., 21, 57-76, (1990) · Zbl 0719.20010
[3] Doerk, K.; Hawkes, T., Finite soluble groups, (1992), de Gruyter Berlin · Zbl 0753.20001
[4] Eick, B.; Wright, C.R.B., Computing subgroups by exhibition in finite solvable groups, J. symb. comput. to appear., (2000)
[5] B. Eick, C. R. B. Wright
[6] The GAP Group
[7] B. Höfling
[8] Holt, D.F.; Rees, S., Testing modules for irreducibility, J. austral. math. soc. ser. A, 57, 1-16, (1994) · Zbl 0833.20021
[9] A. Hulpke, Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation, 1998, Association for Computing Machinery
[10] Laue, R.; Neubüser, J.; Schoenwaelder, U., Algorithms for finite soluble groups and the SOGOS system, () · Zbl 0547.20012
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