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6 Conclusion and Future Work We have shown how different subgroup chains in a permutation group correspond to different Frobenius-Lagrange coordinatizations of that permutation group, as well as to different solving strategies for manipulation. In particular, we showed Computational Understanding and Manipulation of Symmetries 29 that solving strategies for permutation puzzles can be represented by a subgroup chain, which determines a hierarchical decomposition. Coordinatewise manipulation of the permutation group via short or minimal length words over group’s basic generators is an easily achievable next step.

2 Cascade Product of Permutation Groups Given permutation groups (X, G) and (Y, H), their wreath product is the permutation group (X, G) (Y, H) ∼ = (X × Y, G D) Computational Understanding and Manipulation of Symmetries 21 where D = H X is the set of all possible functions from X to H. A state in the wreath product is expressed by two coordinates (x, y), x ∈ X, y ∈ Y . The group elements are coordinatized similarly by (g, d) where g ∈ G and d : X → H is the dependency function, a ‘recipe’ to find an element that should be applied on the second level based on the (previous) state of the first (top) level.

See standard references [2, 27]. e. we can address its parts conveniently and with arbitrary precision. However, we would like to use the coordinate system dynamically, not just as a static description. We would like to calculate with it, finding manipulative operations taking one state to another desired state, or, equivalently, from one tuple of coordinate values to another one using the elementary symmetry operations of the original structure. 1 Component Actions For establishing the connection between the original group and the coordinatized one we need to have a way to express a permutation as coordinate actions.

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