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'' ∈ ''G'' and all ''x'', ''y'' ∈ ''X''. The action of ''G'' on ''X'' induces a natural action of ''G'' on any block system for ''X''.

The set of orbits of the ''G''-set ''X'' is an example of a block system. The corresponding equivalence relation is the smallest ''G''-invariant equivalence on ''X'' such that the induced action on the block system is trivial.

The partition into singleton sets is a block system and if ''X'' is non-empty then the partition into one set ''X'' itself is a block system as well (if ''X'' is a singleton set then these two partitions are identical). A transitive (and thus non-empty) ''G''-set ''X'' is said to be primitive if it has no other block systems. For a non-empty ''G''-set ''X'' the transitivity requirement in the previous definition is only necessary in the case when , ''X'', =''2'' and the group action is trivial.

Characterization of blocks

Each element of some block system is called a block. A block can be characterized as a non-empty subset ''B'' of ''X'' such that for all ''g'' ∈ ''G'', either
*''gB'' = ''B'' (''g'' fixes ''B'') or
*''gB'' ∩ ''B'' = ∅ (''g'' moves ''B'' entirely).
''Proof:'' Assume that ''B'' is a block, and for some ''g'' ∈ ''G'' it's ''gB'' ∩ ''B'' ≠ ∅. Then for some ''x'' ∈ ''B'' it's ''gx'' ~ ''x''. Let ''y'' ∈ ''B'', then ''x'' ~ ''y'' and from the ''G''-invariance it follows that ''gx'' ~ ''gy''. Thus ''y'' ~ ''gy'' and so ''gB'' ⊆ ''B''. The condition ''gx'' ~ ''x'' also implies ''x'' ~ ''g''^−''1''''x'', and by the same method it follows that ''g''^−''1''''B'' ⊆ ''B'', and thus ''B'' ⊆ ''gB''. In the other direction, if the set ''B'' satisfies the given condition then the system together with the complement of the union of these sets is a block system containing ''B''.

In particular, if ''B'' is a block then ''gB'' is a block for any ''g'' ∈ ''G'', and if ''G'' acts transitively on ''X'' then the set is a block system on ''X''.

Stabilizers of blocks

If ''B'' is a block, the stabilizer of ''B'' is the subgroup
:''G''_''B'' = .
The stabilizer of a block contains the stabilizer ''G''_''x'' of each of its elements. Conversely, if ''x'' ∈ ''X'' and ''H'' is a subgroup of ''G'' containing ''G''_''x'', then the orbit ''H''.''x'' of ''x'' under ''H'' is a block contained in the orbit ''G''.''x'' and containing ''x''.

For any ''x'' ∈ ''X'', block ''B'' containing ''x'' and subgroup ''H'' ⊆ ''G'' containing ''G''_''x'' it's ''G''_''B''.''x'' = ''B'' ∩ ''G''.''x'' and ''G''_''H''.''x'' = ''H''.

It follows that the blocks containing ''x'' and contained in ''G''.''x'' are in one-to-one correspondence with the subgroups of ''G'' containing ''G''_''x''. In particular, if the ''G''-set ''X'' is transitive then the blocks containing ''x'' are in one-to-one correspondence with the subgroups of ''G'' containing ''G''_''x''. In this case the ''G''-set ''X'' is primitive if and only if either the group action is trivial (then ''X'' = ) or the stabilizer ''G''_''x'' is a maximal subgroup of ''G'' (then the stabilizers of all elements of ''X'' are the maximal subgroups of ''G'' conjugate to ''G''_''x'' because ''G''_''gx'' = ''g'' ⋅ ''G''_''x'' ⋅ ''g''^−''1'').

See also

*Congruence relation

{{DEFAULTSORT:Block (Group Theory)
Permutation groups