Block (group theory)
Encyclopedia
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
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
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.
The stabilizer of a block contains the stabilizer Gx of each of its elements. Conversely, if x ∈ X and H is a subgroup of G containing Gx, then the orbit of x under H is a block. It follows that the blocks containing x are in one-to-one correspondence with the subgroups of G containing Gx. In particular, a G-set is primitive if and only if the stabilizer of each point is a maximal subgroup
of G.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and group theory
Group theory
In mathematics and 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 as groups endowed with additional operations and...
, a block system for the action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G on a set X is a partition of X that is G-invariant. In terms of the associated equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
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
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
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.
Stabilizers of blocks
If B is a block, the stabilizer of B is the subgroupSubgroup
In group theory, 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 of G if the restriction of * to H x H is a group operation on H...
- GB = { g ∈ G | gB = B }.
The stabilizer of a block contains the stabilizer Gx of each of its elements. Conversely, if x ∈ X and H is a subgroup of G containing Gx, then the orbit of x under H is a block. It follows that the blocks containing x are in one-to-one correspondence with the subgroups of G containing Gx. In particular, a G-set is primitive if and only if the stabilizer of each point is a maximal subgroup
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 strictly. In other words H is a maximal element of the partially...
of G.
See also
- Primitive permutation groupPrimitive permutation groupIn mathematics, a permutation group G acting on a set X is called primitive if G acts transitively on X and G preserves no nontrivial partition of X...
- Congruence relationCongruence relationIn abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure...