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Group action is a concept in which a group acts on a set or structure, with distinct properties and examples such as orbits of fundamental spherical triangles, regular actions of Lie groups on smooth manifolds, and actions of group schemes on algebraic groups.
Key Points
- Group action is a mathematical concept which describes how a group of elements can act on an object of various categories.
- The Orbit-Stabilizer Theorem states that the order of a group's stabilizer is equal to the order of the group, implying that the length of the orbit is a divisor of the group order.
- Examples of well-studied transitive groups are the symmetric group $S_n$, the alternating group $A_n$, and the general linear group $GL(n,V)$.
- A left group action is a bijection from a set of size $2^n$ to the symmetric group, with inverse bijection the corresponding map for right group action.
- The action is called fixed-point free or semiregular (or free) if the statement that no non-trivial element $g = e_G$ fixes a point of $X$ holds true.
- Group actions can be left, right, or linear and have distinct properties and examples.
Summaries
339 word summary
Group actions are functors from a groupoid to a category of sets or another category. Examples include orbits of fundamental spherical triangles, regular actions of Lie groups on smooth manifolds, and actions of group schemes on algebraic groups. Cayley's theorem states that every group is isomorphic to a subgroup of the symmetric group of permutations. Burnside's lemma states that the number of orbits is equal to the average number of points fixed per group element.
The Orbit-Stabilizer Theorem states that the order of a group's stabilizer is equal to the order of the group. As an example, consider a cubical graph with vertices labeled: |G|=8x3x2x1=48. Lie group actions may be irreducible or semisimple and transitive actions have a single orbit. Continuous group actions are homeomorphisms and are properly discontinuous if for every compact subset of the space there are finitely many elements of the group such that their action intersects the subset.
Examples of well-studied transitive groups include the symmetric group $S_n$, the alternating group $A_n$, and the general linear group $GL(n,V)$. The smallest sets on which faithful actions can be defined for these groups vary greatly. For example, the cyclic group $\mathbb{Z}/2^n\mathbb{Z}$ cannot act faithfully on a set of size less than $2n$, while the cyclic group $\mathbb{Z}/2^n\mathbb{Z}$ can act faithfully on a set of size less than $2^n$. Group action is a concept in which a group acts on a set or structure. It can be left, right, or linear and has distinct properties and examples. A right group action is denoted $gx$ and satisfies axioms of compatibility and identity: $\alpha(e,x) = x$ and $\alpha(g, \alpha(h,x)) = \alpha(gh,x)$. Left group action is a bijection from a set of size $2^n$ to the symmetric group, with inverse bijection the corresponding map for right group action. It also satisfies axioms of compatibility and identity: $x \cdot e = x$ and $g \cdot (h \cdot x) = (gh) \cdot x$. An action is transitive when for any two points $x,y\in X$, there exists a $g\in G$ so that $g\cdot x=y$.
660 word summary
Group action is a concept in which a group acts on a space or structure, such as a set of elements, vector space, Euclidean space, or a mathematical structure. Group actions can be left, right, or linear and have distinct properties and examples. The action of a group $G$ on a set $X$ is called transitive if for any two points $x,y\in X$ there exists a $g\in G$ so that $g\cdot x=y$.
A right group action acts first, followed by the second and is often shortened to $gx$. It satisfies the axioms of compatibility and identity, i.e. $\alpha(e,x) = x$ and $\alpha(g, \alpha(h,x)) = \alpha(gh,x)$. A left group action is a bijection from a set of size $2^n$ (of cardinality $(\mathbb{Z}/2\mathbb{Z})^n$) to the symmetric group, with inverse bijection the corresponding map for right group action. It satisfies the axioms of compatibility and identity, i.e. $x \cdot e = x$ and $g \cdot (h \cdot x) = (gh) \cdot x$ respectively.
Examples of well-studied transitive groups are the symmetric group $S_n$, the alternating group $A_n$, and the general linear group $GL(n,V)$. The smallest sets on which faithful actions can be defined for these groups vary greatly. The cyclic group $\mathbb{Z}/2^n\mathbb{Z}$ cannot act faithfully on a set of size less than $2n$, while the cyclic group $\mathbb{Z}/2^n\mathbb{Z}$ can act faithfully on a set of size less than $2^n$.
If the action of $G$ on $X$ is preserved by all elements of $X$, then it is called a principal homogeneous space or -torsor. A continuous group action is an action of a topological group on a topological space by homeomorphisms. It is properly discontinuous if for every compact subset of the space there are finitely many elements of the group such that their action intersects the subset. The action is free and regular if the whole space can be covered by images of a compact subset. The action is continuous for the product topology if a map between the group and the space is continuous.
Lie group actions are smooth on the whole space and may be irreducible or semisimple. The orbit of an element is the set of elements moved by the group, and two elements are equivalent if their orbits are the same. A transitive action has a single orbit, and invariant subsets are those which are fixed under the action.
The Orbit-Stabilizer Theorem states that the order of a group's stabilizer is equal to the order of the group, implying that the length of the orbit is a divisor of the group order. As an example, consider a cubical graph with vertices labeled. Applying the theorem, we can obtain |G|=8x3x2x1=48. Cayley's theorem states that every group is isomorphic to a subgroup of the symmetric group of permutations of a set, and Burnside's lemma states that the number of orbits is equal to the average number of points fixed per group element. The quaternions act as a multiplicative group on versors, the additive group of real numbers acts on the field L, and the Galois group acts on field extensions. The isometries, PGL(2, ), affine group, GL( ), automorphism group, and symmetry group all act on various objects such as 2D images, patterns, projective space, affine space, vector space, and set of points. The symmetric group S acts on its elements and conjugation is an action of every group with subgroup. A semigroup action is defined by using the same two axioms as above.
Group actions are functors from the groupoid to the category of sets or to some other category. Examples include orbit of a fundamental spherical triangle under action of the full icosahedral or octahedral groups, regular actions of smooth manifolds of Lie groups on smooth actions, and actions of group schemes on algebraic groups. Further reading includes Introduction to Abstract Algebra (Smith, 2008), An Introduction to the Theory of Groups (Rotman, 1995), A Course on Abstract Algebra (Eie & Chang, 2010) and Abstract Algebra (Dummit & Foote, 2004).
1480 word summary
Group action is a mathematical concept that describes how a group of elements can act on an object. This action can be applied to objects of various categories, including group objects, schemes, algebraic varieties, smooth manifolds and algebraic groups. Examples include orbit of a fundamental spherical triangle under action of the full icosahedral or octahedral groups, regular actions of smooth manifolds of Lie groups on smooth actions, and actions of group schemes on algebraic groups. Such actions are described by the Transformation Groups (tom Dieck, 1987), Hyperbolic Manifolds and Discrete Groups (Maskit, 1988) and Three-Dimensional Geometry and Topology (Thurston, 1997). Further reading includes Introduction to Abstract Algebra (Smith, 2008), An Introduction to the Theory of Groups (Rotman, 1995), A Course on Abstract Algebra (Eie & Chang, 2010) and Abstract Algebra (Dummit & Foote, 2004). A group action is a functor from the groupoid to the category of sets or to some other category. This notion can be encoded by the groupoid, which has the stabilizers of the action as its vertex groups and orbits of the action as its components.
A morphism between G-sets is a function from one set to another that preserves the structure of the group action. For example, with the quaternions and spatial rotation, a counterclockwise rotation through an angle is a mapping whose elements are sin(/2) and cos(/2 +).
When considering actions on objects of an arbitrary category, a semigroup action is defined by using the same two axioms as above. A morphism between G-sets is bijective if its inverse is also a morphism. Isomorphic G-sets are indistinguishable; for all practical purposes, they are called equivariant maps or G-maps. The quaternions act as a multiplicative group on versors and is useful in studying the action of Mathieu group. The additive group of real numbers acts on the field L and its subgroups correspond to subfields of L, while Galois group acts on field extension. The isometries act on 2D images and patterns, and PGL(2, ) acts on projective space. The affine group acts on affine space, and GL( ) acts on vector space. The automorphism group of vector space and symmetry group of any geometrical object act on set of points. The symmetric group S acts on its elements, and conjugation is an action of every group with subgroup. Cayley's theorem states that every group is isomorphic to a subgroup of the symmetric group of permutations of a set. This action is free and transitive (regular) and forms the basis of a rapid proof of the theorem. Burnside's lemma, a result closely related to the orbit-stabilizer theorem, states that the number of orbits is equal to the average number of points fixed per group element.
As an example, consider a cubical graph with vertices labeled. The group of automorphisms acts on the set of vertices and this action is transitive. By the orbit-stabilizer theorem, we can obtain |G|=8|G_{1}|. Applying the theorem again, we have |G_{1}|=|(G_{1})\cdot 2||(G_{1})_{2}| and |(G_{1})\cdot 2|=3. Finally, any element of G_{1} that fixes 1 must send 2 to either 2, 4, or 5, giving |(G_{1})_{2}|=2. Combining these calculations, we can now obtain |G|=8x3x2x1=48. The Orbit-Stabilizer Theorem states that the order of a group's stabilizer is equal to the order of the group, implying that the length of the orbit is a divisor of the group order. Lagrange's Theorem and Burnside's Lemma follow from this result. For a given group element, the stabilizer subgroup (or isotropy group) is the set of all elements in the group that fix a given element. This induces a bijection between the set of cosets for the stabilizer subgroup and the orbit.
The condition for two elements to have the same image is if they lie in the same coset of the stabilizer subgroup. The action of a group on a set can be represented by a homomorphism with the symmetric group, with kernel equal to the intersection of all stabilizers. If this kernel is trivial, the action is said to be faithful. The higher cohomology groups are derived functors of the functor of G-invariants, and when coefficients are taken into account, the zeroth cohomology group is the set of all such G-invariants. Consider a group G acting on a set X. The orbit of an element x in X is denoted by G.x and is the set of elements in X which can be moved by the elements of G. Two elements x,y in X are equivalent if and only if their orbits are the same, that is, G.x = G.y. The action is transitive if and only if there exists a single orbit, that is, G.x = X. Every orbit is an invariant subset of X, but not conversely. The set of all orbits of X is a partition of X, which is also invariant under G. Every subset that is fixed under G in X (i.e. G.y = y for all g ∈ G) is called an invariant subset.
There is a well-developed theory of Lie group actions, i.e. action which are smooth on the whole space. If G acts by linear transformations on a module over a commutative ring, the action is said to be irreducible if there are no proper nonzero G-invariant submodules. It is said to be semisimple if it decomposes as a direct sum of irreducible actions. Contrary to what the name suggests, this property is weaker than continuity of the action. A continuous group action is an action of a topological group on a topological space by homeomorphisms. For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space. The action is said to be wandering if for every open subset $U\ni x$ there are only finitely many $g\in G$ such that $g\cdot U\cap U\not =\emptyset$. It is also said to be properly discontinuous if for every compact subset $K\subset X$ there are finitely many $g\in G$ such that $g\cdot K\cap K'\not =\emptyset$. The action is said to be free and regular if there exists a compact subset $A\subset X$ such that $X=G\cdot A$. The largest subset on which the action is freely discontinuous is then called the free regular set. The action is continuous for the product topology if the map $(g,x)\mapsto (x,g\cdot x)$ defined by $G\times X\to X\times X$ is continuous. The action of a group $G$ on a set $X$ is called transitive if for any two points $x,y\in X$ there exists a $g\in G$ so that $g\cdot x=y$. This means that $G$ acts simply transitively on $X$ (or sharply transitive, or multiply transitive) if it is both transitive and free. If the action of $G$ on $X$ is preserved by all elements of $X$, then it is called a principal homogeneous space or -torsor.
Examples of well-studied transitive groups are the symmetric group $S_n$, the alternating group $A_n$, and the general linear group $GL(n,V)$. The smallest sets on which faithful actions can be defined for these groups can vary greatly. For example, three groups of size 120 are the symmetric group $S_5$, the icosahedral group $A_5\times \mathbb{Z}/2\mathbb{Z}$, and the cyclic group $\mathbb{Z}/120\mathbb{Z}$. The cyclic group $\mathbb{Z}/2^n\mathbb{Z}$ cannot act faithfully on a set of size less than $2n$, while the cyclic group $\mathbb{Z}/2^n\mathbb{Z}$ can act faithfully on a set of size less than $2^n$. A group can be embedded in a symmetric group, which is infinite when the group is finite. Thus, for general properties of group actions, it suffices to consider only left actions. However, there are cases where this is not possible, e.g. when considering the multiplication of a group on itself by left and right multiplication.
A left group action is a bijection from a set of size $2^n$ (of cardinality $(\mathbb{Z}/2\mathbb{Z})^n$) to the symmetric group, with inverse bijection the corresponding map for right group action. It satisfies the axioms of compatibility and identity, i.e. $x \cdot e = x$ and $g \cdot (h \cdot x) = (gh) \cdot x$ respectively.
A right group action acts first, followed by the second and is often shortened to $gx$. It satisfies the axioms of compatibility and identity, i.e. $\alpha(e,x) = x$ and $\alpha(g, \alpha(h,x)) = \alpha(gh,x)$.
The action is called fixed-point free or semiregular (or free) if the statement that no non-trivial element $g = e_G$ fixes a point of $X$ holds true. This is a much stronger property than faithfulness. Group action is a mathematical concept in which a group acts on a space or structure, such as a set of elements, vector space, Euclidean space, or a mathematical structure. Group actions can be left, right, or linear and have distinct properties and examples. The group of symmetries of a polyhedron acts on the set of its vertices, edges, and faces. Similarly, the group of Euclidean isometries acts on Euclidean space and its objects. Group actions are also represented by group homomorphisms of a group into the automorphism group of the space.