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Minimal and canonical images. (English) Zbl 1439.20001
Assume \(G\) acts on a set \(\Omega\) and \(X\) is a subset of \(\Omega\). Let \(\Omega_1, \ldots,\Omega_n\) be the orbits of \(G\) on \(\Omega\). The problem is to write \(X\) as the union of \(\Omega_i \cap X\). The authors of this paper offer an algorithm to solve this problem. The idea is to use a so called canonical labelling function which gives a canonical image for each element in \(\Omega\). The purpose of the algorithm is to find the canonical image. Afterwards one can put the canonical image into equivalence classes. The canonical image problem goes back to J. S. Leon [J. Symb. Comput. 12, No. 4–5, 533–583 (1991; Zbl 0807.20001)] one of the pioneers of computational group theory. In this paper the authors prove correctness of the algorithm and compare it with existing algorithms.

20-08 Computational methods for problems pertaining to group theory
20B99 Permutation groups
Ferret; GAP; images; nauty; Traces
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