Structure tree

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

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(I will also put a diagram to make things clearer).