TOPICS

# 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 generalizes that of a minimal group block.

Fundamental Domain, Group, Primitive Group Action, Steiner System

Portions of this entry contributed by Todd Rowland

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## Cite this as:

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