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.

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See also

See also

★ Primitive permutation group

★ Congruence relation