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 subsetB 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−1x, and by the same method it follows that g−1B ⊆ B, and thus B ⊆ gB. In the other direction, if the set B satisfies the given condition then the system {gB | g ∈ G} 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 {gB | g ∈ G} is a block system on X.
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 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 Gx it's GB.x = B ∩ G.x and GH.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 Gx. 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 Gx. In this case the G-set X is primitive if and only if either the group action is trivial (then X = {x}) or the stabilizer Gx is a maximal subgroup of G (then the stabilizers of all elements of X are the maximal subgroups of Gconjugate to Gx because Ggx = g ⋅ Gx ⋅ g−1).
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c:
In mathematics, when the elements of some set S have a notion of equivalence defined on them, then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if and only if they are equivalent.
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. One says that the group acts on the space or structure. If a group acts on a structure, it also acts on everything that is built on the structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. In particular, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron.
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cases of a preorder. An antisymmetric preorder is a partial order, and a symmetric preorder is an equivalence relation.
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group Sn defined over a finite set of n symbols consists of the permutation operations that can be performed on the n symbols. Since there are n! such permutation operations, the order of the symmetric group Sn is n!.
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = g–1ag. This is an equivalence relation whose equivalence classes are called conjugation classes.
In mathematics, a partition of a set is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in exactly one subset.
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements of G are called the symmetries of X. A special case of this is when the group G in question is the automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, X is homogeneous if intuitively X looks locally the same at each point, either in the sense of isometry, diffeomorphism, or homeomorphism (topology). Some authors insist that the action of G be faithful, although the present article does not. Thus there is a group action of G on X which can be thought of as preserving some "geometric structure" on X, and making X into a single G-orbit.
In group theory, a field of mathematics, a double coset is a collection of group elements which are equivalent under the symmetries coming from two subgroups. More precisely, let G be a group, and let H and K be subgroups. Let H act on G by left multiplication while K acts on G by right multiplication. For each x in G, the (H, K)-double coset of x is the set
The concept of system of imprimitivity is used in mathematics, particularly in algebra and analysis, both within the context of the theory of group representations. It was used by George Mackey as the basis for his theory of induced unitary representations of locally compact groups.
A one-dimensional symmetry group is a mathematical group that describes symmetries in one dimension (1D).
In mathematics, D3 (sometimes also denoted by D6) is the dihedral group of degree 3, which is isomorphic to the symmetric group S3 of degree 3. It is also the smallest possible non-abelian group.
In mathematics, a permutation group G acting on a non-empty set X is called primitive if G acts transitively on X and G preserves no nontrivial partition of X, where nontrivial partition means a partition that isn't a partition into singleton sets or partition into one set X. Otherwise, if G is transitive and G does preserve a nontrivial partition, G is called imprimitive.
In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has as an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds.
In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on , functions on a manifold, vector valued functions, vector fields, or, more generally, sections of a vector bundle.
In the area of modern algebra known as group theory, the Mathieu groupM24 is a sporadic simple group of order
A 2-transitive group is a transitive group used in group theory in which the stabilizer subgroup of every point acts transitively on the remaining points. Equivalently, a group acts 2-transitively on a set if it acts transitively on the set of distinct ordered pairs . That is, assuming that acts on the left of , for each pair of pairs with and , there exists a such that . Equivalently, and , since the induced action on the distinct set of pairs is .
In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a rational root if and only if the Galois group of p is included in G. More exactly, if the Galois group is included in G, then the resolvent has a rational root, and the converse is true if the rational root is a simple root. Resolvents were introduced by Joseph Louis Lagrange and systematically used by Évariste Galois. Nowadays they are still a fundamental tool to compute Galois groups. The simplest examples of resolvents are
In mathematics, Borel–de Siebenthal theory describes the closed connected subgroups of a compact Lie group that have maximal rank, i.e. contain a maximal torus. It is named after the Swiss mathematicians Armand Borel and Jean de Siebenthal who developed the theory in 1949. Each such subgroup is the identity component of the centralizer of its center. They can be described recursively in terms of the associated root system of the group. The subgroups for which the corresponding homogeneous space has an invariant complex structure correspond to parabolic subgroups in the complexification of the compact Lie group, a reductive algebraic group.
In mathematics, an invariant convex cone is a closed convex cone in a Lie algebra of a connected Lie group that is invariant under inner automorphisms. The study of such cones was initiated by Ernest Vinberg and Bertram Kostant.
This page is based on this Wikipedia article Text is available under the CC BY-SA 4.0 license; additional terms may apply. Images, videos and audio are available under their respective licenses.
