Group Block

A group action G×Omega->Omega might preserve a special kind of partition of Omega called a system of blocks. A block is a subset Delta of Omega such that for any group element g either

1. g preserves Delta, i.e., gDelta=Delta, or

2. g translates everything in Delta out of Delta, i.e., gDelta intersection Delta=phi.

For example, the general linear group GL(2,R) acts on the plane minus the origin, R^2-(0,0). The lines A={(at,bt)} 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 A are minimal blocks.

In fact, if two blocks intersect then their intersection is also a block. Hence, the minimal blocks form a partition of Omega. 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.

See also

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.

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