A group block for a group action 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 punctured lines , where , are blocks because either a line is mapped to itself, or to another line. These punctured lines are not minimal blocks. Every singleton is a block, and each punctured line contains positive rays, which are also blocks.

In fact, if two blocks intersect then their intersection is also a block. Hence, the minimal nonempty blocks under inclusion are singletons, forming the singleton set partition of . It is important to avoid confusion with the notion of a block in a block design, which is different.

A fundamental domain is instead defined by the orbits of the group action.

See also

Block, Fundamental Domain, Group, Primitive Group Action, Steiner System

Portions of this entry contributed by Todd Rowland

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Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Group Block." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GroupBlock.html

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