Block (permutation group theory)
In mathematics and group theory, a
block system for the action of a group
G on a set
X is a partition of
X that is
G-invariant. In terms of the associated equivalence relation on
X,
G-invariance means that
x ~
y implies
gx ~
gy for all
g in
G and all
x,
y in
X. The action of
G on
X determines a natural action of
G on any block system for
X. Each element of the block system is called a
block. A block can be characterized as a subset
B of
X such that for all
g in
G, either ▪
gB =
B or ▪
gB ∩
B = ∅. If
B is a block then
gB is a block for any
g in
G. If
G acts transitively on
X, then the set is a block system on
X. The trivial partitions into singleton sets and the partition into one set
X itself are block systems. A transitive
G-set
X is said to be
primitive if contains no nontrivial partitions.