mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and

group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...

, a block system for the action of a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

''G'' on a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

''X'' is a partition of ''X'' that is ''G''-invariant. In terms of the associated

equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...

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 set In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the ...

s 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 In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...

''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 In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a 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 In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...

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 In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra. In group theory, a maximal subgroup ''H'' of a group ''G'' is a proper subgroup, such that no proper subgroup ''K'' contains ''H'' ...

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'').

Characterization of blocks

Stabilizers of blocks

See also