Structure tree

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Definition

Definition with symbols

Let G be a group acting transitively on a set S. A structure tree for the action of G on S is a tree described as follows:

  • Put all the elements of S at the leaves of the tree
  • Find a minimal nontrivial block decomposition for S, or equivalently, a decomposition into blocks such that each block contains no proper nontrivial block. Represent each block by a point, and make this set of points form the layer of the tree just above the leaves, such that the children of each block are the elements of that block.
  • Now, G acts on the set of blocks. Treat this as the new set, again find a minaml block decomposition and make the next layer.
  • Keep repeating this process till the action of G on the blocks becomes primitive, in which case the next layer will be the root (The block representing the whole of G)

Example

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] (I will also put a diagram to make things clearer).