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.