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Group block

A group action might preserve a special kind of partition of called a system of blocks. A block is a subset of such that for any group element either 1. preserves , i.e., , or 2. translates everything in out of , i.e., . For example, the general linear group acts on the plane minus the origin, . The lines are blocks because either a line is mapped to itself, or to another line. Of course, the points on the line may be rescaled, so the lines in are minimal blocks.In fact, if two blocks intersect then their intersection is also a block. Hence, the minimal blocks form a partition of . It is important to avoid confusion with the notion of a block in a block design, which is different.The concept of a fundamental domain generalizesthat of a minimal group block.

Fundamental domain

Let be a group and be a topological G-set. Then a closed subset of is called a fundamental domain of in if is the union of conjugates of , i.e.,and the intersection of any two conjugates has no interior.For example, a fundamental domain of the group of rotations by multiples of in is the upper half-plane and a fundamental domain of rotations by multiples of is the first quadrant .The concept of a fundamental domain is a generalization of a minimal group block, since while the intersection of fundamental domains has empty interior, the intersection of minimal blocks is the empty set.

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