Block (group theory)

Block (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" ("g" fixes "B") or
*"gB" ∩ "B" = ∅ ("g" moves "B" entirely).If "B" is a block then "gB" is a block for any "g" in "G". If "G" acts transitively on "X", then the set {"gB" | "g" ∈ "G"} 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.

See also

*Primitive permutation group
*Congruence relation


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