Group action

In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space. Common examples of spaces that groups act on are sets, vector spaces, and topological spaces. Actions of groups on vector spaces are called representations of the group.

When there is a natural correspondence between the set of group elements and the set of space transformations, a group can be interpreted as acting on the space in a canonical way. For example, the symmetric group of a finite set consists of all bijective transformations of that set; thus, applying any element of the permutation group to an element of the set will produce another (not necessarily distinct) element of the set. More generally, symmetry groups such as the homeomorphism group of a topological space or the general linear group of a vector space, as well as their subgroups, also admit canonical actions. For other groups, an interpretation of the group in terms of an action may have to be specified, either because the group does not act canonically on any space or because the canonical action is not the action of interest. For example, we can specify an action of the two-element cyclic group on the finite set by specifying that 0 (the identity element) sends , and that 1 sends . This action is not canonical.

A common way of specifying non-canonical actions is to describe a homomorphism from a group G to the group of symmetries of a set X. The action of an element on a point is assumed to be identical to the action of its image on the point . The homomorphism is also frequently called the "action" of G, since specifying is equivalent to specifying an action. Thus, if G is a group and X is a set, then an action of G on X may be formally defined as a group homomorphism from G to the symmetric group of X. The action assigns a permutation of X to each element of the group in such a way that:

If X has additional structure, then is only called an action if for each , the permutation preserves the structure of X.

The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them. In particular, groups can act on other groups, or even on themselves. Because of this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields.

Group action on equilateral triangle
Given an equilateral triangle, the counterclockwise rotation by 120° around the center of the triangle maps every vertex of the triangle to another one. The cyclic group C3 consisting of the rotations by 0°, 120° and 240° acts on the set of the three vertices.

Definition

Left Group Action

If G is a group and X is a set, then a (left) group action φ of G on X is a function

that satisfies the following two axioms (where we denote φ(g, x) as gx):[1]

Identity
ex = x for all x in X. (Here, e denotes the identity element of the group G.)
Compatibility
(gh)⋅x = g⋅(hx) for all g, h in G and all x in X. (Here, gh denotes the result of applying the group operation of G to the elements g and h.)

The group G is said to act on X (on the left). The set X is called a (left) G-set.

From these two axioms, it follows that for every g in G, the function which maps x in X to gx is a bijective map from X to X (its inverse being the function which maps x to g−1x). Therefore, one may alternatively define a group action of G on X as a group homomorphism from G into the symmetric group Sym(X) of all bijections from X to X.[2]

Right Group Action

In complete analogy, one can define a right group action of G on X as an operation X × GX mapping (x, g) to xg and satisfying the two axioms:

Identity
xe = x for all x in X.
Compatibility
x⋅(gh) = (xg)⋅h for all g, h in G and all x in X;

The difference between left and right actions is in the order in which a product like gh acts on x. For a left action h acts first and is followed by g, while for a right action g acts first and is followed by h. Because of the formula (gh)−1 = h−1g−1, one can construct a left action from a right action by composing with the inverse operation of the group. Also, a right action of a group G on X is the same thing as a left action of its opposite group Gop on X. It is thus sufficient to only consider left actions without any loss of generality.

Types of actions

The action of G on X is called

  • Transitive if X is non-empty and if for each pair x, y in X there exists a g in G such that gx = y. For example, the action of the symmetric group of X is transitive, the action of the general linear group or the special linear group of a vector space V on V ∖ {0} is transitive, but the action of the orthogonal group of a Euclidean space E is not transitive on E ∖ {0} (it is transitive on the unit sphere of E, though).
  • Faithful (or effective) if for every two distinct g, h in G there exists an x in X such that gxhx; or equivalently, if for each ge in G there exists an x in X such that gxx. In other words, in a faithful group action, different elements of G induce different permutations of X.[a] In algebraic terms, a group G acts faithfully on X if and only if the corresponding homomorphism to the symmetric group, G → Sym(X), has a trivial kernel. Thus, for a faithful action, G embeds into a permutation group on X; specifically, G is isomorphic to its image in Sym(X). If G does not act faithfully on X, one can easily modify the group to obtain a faithful action. If we define N = {g in G : gx = x for all x in X}, then N is a normal subgroup of G; indeed, it is the kernel of the homomorphism G → Sym(X). The factor group G/N acts faithfully on X by setting (gN)⋅x = gx. The original action of G on X is faithful if and only if N = {e}.
  • Free (or semiregular or fixed point free) if, given g, h in G, the existence of an x in X with gx = hx implies g = h. Equivalently: if g is a group element and there exists an x in X with gx = x (that is, if g has at least one fixed point), then g is the identity. Note that a free action on a non-empty set is faithful.
  • Regular (or simply transitive or sharply transitive) if it is both transitive and free; this is equivalent to saying that for every two x, y in X there exists precisely one g in G such that gx = y. In this case, X is called a principal homogeneous space for G or a G-torsor. The action of any group G on itself by left multiplication is regular, and thus faithful as well. Every group can, therefore, be embedded in the symmetric group on its own elements, Sym(G). This result is known as Cayley's theorem.
  • n-transitive if X has at least n elements and for all pairwise distinct x1, ..., xn and pairwise distinct y1, ..., yn there is a g in G such that gxk = yk for 1 ≤ kn. A 2-transitive action is also called doubly transitive, a 3-transitive action is also called triply transitive, and so on. Such actions define interesting classes of subgroups in the symmetric groups: 2-transitive groups and more generally multiply transitive groups. The action of the symmetric group on a set with n elements is always n-transitive; the action of the alternating group is n-2-transitive.
  • Sharply n-transitive if there is exactly one such g.
  • Primitive if it is transitive and preserves no non-trivial partition of X. See primitive permutation group for details.
  • Locally free if G is a topological group, and there is a neighbourhood U of e in G such that the restriction of the action to U is free; that is, if gx = x for some x and some g in U then g = e.

Furthermore, if acts on a topological space , then the action is:

  • Wandering if every point has a neighbourhood such that is finite.[3] For example, the action of on by translations is wandering. The action of the modular group on the Poincaré half-plane is also wandering.
  • Properly discontinuous if is a locally compact space and for every compact subset the set is finite. The wandering actions given above are also properly discontinuous. On the other hand, the action of on by the linear map is wandering and free but not properly discontinuous.[4]
  • Proper if is a topological group and the map from is proper.[5] If G is discrete then properness is equivalent to proper discontinuity for G-actions.
  • Said to have discrete orbits if the orbit of each under the action of is discrete in .[3]

If X is a non-zero module over a ring R and the action of G is R-linear then it is said to be

  • Irreducible if there is no nonzero proper invariant submodule.

Orbits and stabilizers

Compound of five tetrahedra
In the compound of five tetrahedra, the symmetry group is the (rotational) icosahedral group I of order 60, while the stabilizer of a single chosen tetrahedron is the (rotational) tetrahedral group T of order 12, and the orbit space I/T (of order 60/12 = 5) is naturally identified with the 5 tetrahedra – the coset gT corresponds to the tetrahedron to which g sends the chosen tetrahedron.

Consider a group G acting on a set X. The orbit of an element x in X is the set of elements in X to which x can be moved by the elements of G. The orbit of x is denoted by Gx:

The defining properties of a group guarantee that the set of orbits of (points x in) X under the action of G form a partition of X. The associated equivalence relation is defined by saying xy if and only if there exists a g in G with gx = y. The orbits are then the equivalence classes under this relation; two elements x and y are equivalent if and only if their orbits are the same; i.e., Gx = Gy.

The group action is transitive if and only if it has exactly one orbit, i.e., if there exists x in X with Gx = X. This is the case if and only if Gx = X for all x in X.

The set of all orbits of X under the action of G is written as X/G (or, less frequently: G\X), and is called the quotient of the action. In geometric situations it may be called the orbit space, while in algebraic situations it may be called the space of coinvariants, and written XG, by contrast with the invariants (fixed points), denoted XG: the coinvariants are a quotient while the invariants are a subset. The coinvariant terminology and notation are used particularly in group cohomology and group homology, which use the same superscript/subscript convention.

Invariant subsets

If Y is a subset of X, we write GY for the set { gy : yY and gG}. We call the subset Y invariant under G if GY = Y (which is equivalent to GYY). In that case, G also operates on Y by restricting the action to Y. The subset Y is called fixed under G if gy = y for all g in G and all y in Y. Every subset that is fixed under G is also invariant under G, but not vice versa.

Every orbit is an invariant subset of X on which G acts transitively. The action of G on X is transitive if and only if all elements are equivalent, meaning that there is only one orbit.

A G-invariant element of X is xX such that gx = x for all gG. The set of all such x is denoted XG and called the G-invariants of X. When X is a G-module, XG is the zeroth group cohomology group of G with coefficients in X, and the higher cohomology groups are the derived functors of the functor of G-invariants.

Fixed points and stabilizer subgroups

Given g in G and x in X with gx = x, we say x is a fixed point of g and g fixes x.

For every x in X, we define the stabilizer subgroup of G with respect to x (also called the isotropy group or little group[6]) as the set of all elements in G that fix x:

This is a subgroup of G, though typically not a normal one. The action of G on X is free if and only if all stabilizers are trivial. The kernel N of the homomorphism with the symmetric group, G → Sym(X), is given by the intersection of the stabilizers Gx for all x in X. If N is trivial, the action is said to be faithful (or effective).

Let x and y be two elements in X, and let g be a group element such that y = gx. Then the two stabilizer groups Gx and Gy are related by Gy = g Gx g−1. Proof: by definition, hGy if and only if h⋅(gx) = gx. Applying g−1 to both sides of this equality yields (g−1hg)⋅x = x; that is, g−1hgGx. An opposite inclusion follows similarly by taking hGx and supposing x = g−1y.

The above says that the stabilizers of elements in the same orbit are conjugate to each other. Thus, to each orbit, one can associate a conjugacy class of a subgroup of G (i.e., the set of all conjugates of the subgroup). Let denote the conjugacy class of H. Then one says that the orbit O has type if the stabilizer of some/any x in O belongs to . A maximal orbit type is often called a principal orbit type.

Orbit-stabilizer theorem and Burnside's lemma

Orbits and stabilizers are closely related. For a fixed , consider the map given by . This map induces a bijection from the set of cosets of in to the orbit , as the translate depends only on the left coset [7]. This result is known as the orbit-stabilizer theorem.

If is finite then the orbit-stabilizer theorem, together with Lagrange's theorem, gives

This result is especially useful since it can be employed for counting arguments (typically in situations where is finite as well).

Labeled cube graph
Cubical graph with vertices labeled
Example: One can use the orbit-stabilizer theorem to count the automorphisms of a graph. Consider the cubical graph as pictured, and let denote its automorphism group. Then acts on the set of vertices , and this action is transitive as can be seen by composing rotations about the center of the cube. Thus, by the orbit-stabilizer theorem, we have that . Applying the theorem now to the stabilizer , we obtain . Any element of that fixes must send to either , or . There are such automorphisms; consider for example the map that transposes and , transposes and , and fixes the other vertices. Thus, . Applying the theorem a third time gives . Any element of that fixes and must send to either or , and one easily finds such automorphisms. Thus, . One also sees that consists only of the identity automorphism, as any element of fixing , and must also fix and consequently all other vertices. Combining the preceding calculations, we now obtain .

A result closely related to the orbit-stabilizer theorem is Burnside's lemma:

,

where the set of points fixed by . This result is mainly of use when and are finite, when it can be interpreted as follows: the number of orbits is equal to the average number of points fixed per group element.

Fixing a group , the set of formal differences of finite -sets forms a ring called the Burnside ring of , where addition corresponds to disjoint union, and multiplication to Cartesian product.

Examples

  • The trivial action of any group G on any set X is defined by gx = x for all g in G and all x in X; that is, every group element induces the identity permutation on X.[8]
  • In every group G, left multiplication is an action of G on G: gx = gx for all g, x in G. This action forms the basis of a rapid proof of Cayley's theorem - that every group is isomorphic to a subgroup of the symmetric group of permutations of the set G.
  • In every group G with subgroup H, left multiplication is an action of G on the set of cosets G/H: gaH = gaH for all g,a in G. In particular if H contains no nontrivial normal subgroups of G this induces an isomorphism from G to a subgroup of the permutation group of degree [G : H].
  • In every group G, conjugation is an action of G on G: gx = gxg−1. An exponential notation is commonly used for the right-action variant: xg = g−1xg; it satisfies (xg)h = xgh.
  • In every group G with subgroup H, conjugation is an action of G on conjugates of H: gK = gKg−1 for all g in G and K conjugates of H.
  • The symmetric group Sn and its subgroups act on the set { 1, …, n } by permuting its elements
  • The symmetry group of a polyhedron acts on the set of vertices of that polyhedron. It also acts on the set of faces or the set of edges of the polyhedron.
  • The symmetry group of any geometrical object acts on the set of points of that object.
  • The automorphism group of a vector space (or graph, or group, or ring…) acts on the vector space (or set of vertices of the graph, or group, or ring…).
  • The general linear group GL(n, K) and its subgroups, particularly its Lie subgroups (including the special linear group SL(n, K), orthogonal group O(n, K), special orthogonal group SO(n, K), and symplectic group Sp(n, K)) are Lie groups that act on the vector space Kn. The group operations are given by multiplying the matrices from the groups with the vectors from Kn.
  • The general linear group GL(n, Z) acts on Zn by natural matrix action. The orbits of its action are classified by the greatest common divisor of coordinates of the vector in Zn.
  • The affine group acts transitively on the points of an affine space, and the subgroup V of the affine group (i.e., a vector space) transitive and free (i.e., regular) action on these points;[9] indeed this can be used to give a definition of an affine space.
  • The projective linear group PGL(n + 1, K) and its subgroups, particularly its Lie subgroups, which are Lie groups that act on the projective space Pn(K). This is a quotient of the action of the general linear group on projective space. Particularly notable is PGL(2, K), the symmetries of the projective line, which is sharply 3-transitive, preserving the cross ratio; the Möbius group PGL(2, C) is of particular interest.
  • The isometries of the plane act on the set of 2D images and patterns, such as wallpaper patterns. The definition can be made more precise by specifying what is meant by image or pattern; e.g., a function of position with values in a set of colors. Isometries are in fact one example of affine group (action).
  • The sets acted on by a group G comprise the category of G-sets in which the objects are G-sets and the morphisms are G-set homomorphisms: functions f : XY such that g⋅(f(x)) = f(gx) for every g in G.
  • The Galois group of a field extension L/K acts on the field L but has only a trivial action on elements of the subfield K. Subgroups of Gal(L/K) correspond to subfields of L that contain K, i.e., intermediate field extensions between L and K.
  • The additive group of the real numbers (R, +) acts on the phase space of "well-behaved" systems in classical mechanics (and in more general dynamical systems) by time translation: if t is in R and x is in the phase space, then x describes a state of the system, and t + x is defined to be the state of the system t seconds later if t is positive or −t seconds ago if t is negative.
  • The additive group of the real numbers (R, +) acts on the set of real functions of a real variable in various ways, with (tf)(x) equal to, e.g., f(x + t), f(x) + t, f(xet), f(x)et, f(x + t)et, or f(xet) + t, but not f(xet + t).
  • Given a group action of G on X, we can define an induced action of G on the power set of X, by setting gU = {gu : uU} for every subset U of X and every g in G. This is useful, for instance, in studying the action of the large Mathieu group on a 24-set and in studying symmetry in certain models of finite geometries.
  • The quaternions with norm 1 (the versors), as a multiplicative group, act on R3: for any such quaternion z = cos α/2 + v sin α/2, the mapping f(x) = zxz is a counterclockwise rotation through an angle α about an axis given by a unit vector v; z is the same rotation; see quaternions and spatial rotation.

Group actions and groupoids

The notion of group action can be put in a broader context by using the action groupoid associated to the group action, thus allowing techniques from groupoid theory such as presentations and fibrations. Further the stabilizers of the action are the vertex groups, and the orbits of the action are the components, of the action groupoid. For more details, see the book Topology and groupoids referenced below.

This action groupoid comes with a morphism which is a covering morphism of groupoids. This allows a relation between such morphisms and covering maps in topology.

Morphisms and isomorphisms between G-sets

If X and Y are two G-sets, we define a morphism from X to Y to be a function f : XY such that f(gx) = gf(x) for all g in G and all x in X. Morphisms of G-sets are also called equivariant maps or G-maps.

The composition of two morphisms is again a morphism.

If a morphism f is bijective, then its inverse is also a morphism, and we call f an isomorphism and the two G-sets X and Y are called isomorphic; for all practical purposes, they are indistinguishable in this case.

Some example isomorphisms:

  • Every regular G action is isomorphic to the action of G on G given by left multiplication.
  • Every free G action is isomorphic to G × S, where S is some set and G acts on G × S by left multiplication on the first coordinate. (S can be taken to be the set of orbits X/G.)
  • Every transitive G action is isomorphic to left multiplication by G on the set of left cosets of some subgroup H of G. (H can be taken to be the stabilizer group of any element of the original G-set.the original action.)

With this notion of morphism, the collection of all G-sets forms a category; this category is a Grothendieck topos (in fact, assuming a classical metalogic, this topos will even be Boolean).

Continuous group actions

One often considers continuous group actions: the group G is a topological group, X is a topological space, and the map G × XX is continuous with respect to the product topology of G × X. The space X is also called a G-space in this case. This is indeed a generalization, since every group can be considered a topological group by using the discrete topology. All the concepts introduced above still work in this context, however we define morphisms between G-spaces to be continuous maps compatible with the action of G. The quotient X/G inherits the quotient topology from X, and is called the quotient space of the action. The above statements about isomorphisms for regular, free and transitive actions are no longer valid for continuous group actions.

If X is a regular covering space of another topological space Y, then the action of the deck transformation group on X is properly discontinuous as well as being free. Every free, properly discontinuous action of a group G on a path-connected topological space X arises in this manner: the quotient map XX/G is a regular covering map, and the deck transformation group is the given action of G on X. Furthermore, if X is simply connected, the fundamental group of X/G will be isomorphic to G.

These results have been generalized in the book Topology and Groupoids referenced below to obtain the fundamental groupoid of the orbit space of a discontinuous action of a discrete group on a Hausdorff space, as, under reasonable local conditions, the orbit groupoid of the fundamental groupoid of the space. This allows calculations such as the fundamental group of the symmetric square of a space X, namely the orbit space of the product of X with itself under the twist action of the cyclic group of order 2 sending (x, y) to (y, x).

An action of a group G on a locally compact space X is cocompact if there exists a compact subset A of X such that GA = X. For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space X/G.

The action of G on X is said to be proper if the mapping G × XX × X that sends (g, x) ↦ (g⋅x, x) is a proper map.

Strongly continuous group action and smooth points

A group action of a topological group G on a topological space X is said to be strongly continuous if for all x in X, the map ggx is continuous with respect to the respective topologies. Such an action induces an action on the space of continuous functions on X by defining (gf)(x) = f(g−1x) for every g in G, f a continuous function on X, and x in X. Note that, while every continuous group action is strongly continuous, the converse is not in general true.[10]

The subspace of smooth points for the action is the subspace of X of points x such that ggx is smooth; i.e., it is continuous and all derivatives are continuous.

Variants and generalizations

One can also consider actions of monoids on sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however. See semigroup action.

Instead of actions on sets, one can define actions of groups and monoids on objects of an arbitrary category: start with an object X of some category, and then define an action on X as a monoid homomorphism into the monoid of endomorphisms of X. If X has an underlying set, then all definitions and facts stated above can be carried over. For example, if we take the category of vector spaces, we obtain group representations in this fashion.

One can view a group G as a category with a single object in which every morphism is invertible. A (left) group action is then nothing but a (covariant) functor from G to the category of sets, and a group representation is a functor from G to the category of vector spaces. A morphism between G-sets is then a natural transformation between the group action functors. In analogy, an action of a groupoid is a functor from the groupoid to the category of sets or to some other category.

In addition to continuous actions of topological groups on topological spaces, one also often considers smooth actions of Lie groups on smooth manifolds, regular actions of algebraic groups on algebraic varieties, and actions of group schemes on schemes. All of these are examples of group objects acting on objects of their respective category.

See also

Notes

  1. ^ i.e. the associated permutation representation is injective.

Citations

  1. ^ Eie & Chang (2010). A Course on Abstract Algebra. p. 144.
  2. ^ This is done, e.g., by Smith (2008). Introduction to abstract algebra. p. 253.
  3. ^ a b Thurston, William (1980), The geometry and topology of three-manifolds, Princeton lecture notes, p. 175
  4. ^ Thurston 1980, p. 176.
  5. ^ tom Dieck, Tammo (1987), Transformation groups, de Gruyter Studies in Mathematics, 8, Berlin: Walter de Gruyter & Co., p. 29, doi:10.1515/9783110858372.312, ISBN 978-3-11-009745-0, MR 0889050
  6. ^ Procesi, Claudio (2007). Lie Groups: An Approach through Invariants and Representations. Springer Science & Business Media. p. 5. ISBN 9780387289298. Retrieved 23 February 2017.
  7. ^ M. Artin, Algebra, Proposition 6.4 on p. 179
  8. ^ Eie & Chang (2010). A Course on Abstract Algebra. p. 145.
  9. ^ Reid, Miles (2005). Geometry and topology. Cambridge, UK New York: Cambridge University Press. p. 170. ISBN 9780521613255.
  10. ^ Yuan, Qiaochu (27 February 2013). "wiki's definition of "strongly continuous group action" wrong?". Mathematics Stack Exchange. Retrieved 1 April 2013.

References

  • Aschbacher, Michael (2000). Finite Group Theory. Cambridge University Press. ISBN 978-0-521-78675-1. MR 1777008.
  • Brown, Ronald (2006). Topology and groupoids, Booksurge PLC, ISBN 1-4196-2722-8.
  • Categories and groupoids, P.J. Higgins, downloadable reprint of van Nostrand Notes in Mathematics, 1971, which deal with applications of groupoids in group theory and topology.
  • Dummit, David; Richard Foote (2004). Abstract Algebra (3rd ed.). Wiley. ISBN 0-471-43334-9.
  • Eie, Minking; Chang, Shou-Te (2010). A Course on Abstract Algebra. World Scientific. ISBN 978-981-4271-88-2.
  • Rotman, Joseph (1995). An Introduction to the Theory of Groups. Graduate Texts in Mathematics 148 (4th ed.). Springer-Verlag. ISBN 0-387-94285-8.
  • Smith, Jonathan D.H. (2008). Introduction to abstract algebra. Textbooks in mathematics. CRC Press. ISBN 978-1-4200-6371-4.

External links

Action

Action may refer to:

Action (narrative), a literary mode

Action fiction, a type of genre fiction

Action game, a genre of video game

Cocompact group action

In mathematics, an action of a group G on a topological space X is cocompact if the quotient space X/G is a compact space. If X is locally compact, then an equivalent condition is that there is a compact subset K of X such that the image of K under the action of G covers X. It is sometimes referred to as mpact, a tongue-in-cheek reference to dual notions where prefixing with "co-" twice would "cancel out".

Conjugacy class

In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure. For an abelian group, each conjugacy class is a set containing one element (singleton set).

Functions that are constant for members of the same conjugacy class are called class functions.

Equivalence class

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) 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 a and b are equivalent.

Formally, given a set S and an equivalence relation ~ on S, the equivalence class of an element a in S is the set

of elements which are equivalent to a. It may be proven from the defining properties of "equivalence relations" that the equivalence classes form a partition of S. This partition – the set of equivalence classes – is sometimes called the quotient set or the quotient space of S by ~ and is denoted by S / ~.

When the set S has some structure (such as a group operation or a topology) and the equivalence relation ~ is defined in a manner suitably compatible with this structure, then the quotient set often inherits a similar structure from its parent set. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories.

Fundamental domain

Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits.

There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action then tile the space. One general construction of fundamental domains uses Voronoi cells.

G.I. Joe

G.I. Joe is a line of action figures produced and owned by the toy company Hasbro. The initial product offering represented four of the branches of the U.S. armed forces with the Action Soldier (U.S. Army), Action Sailor (U.S. Navy), Action Pilot (Air Force), Action Marine (Marine Corps) and later on, the Action Nurse. The name derived from the usage of "G.I. Joe" for the generic U.S. soldier, itself derived from the more general term "G.I.". The development of G.I. Joe led to the coining of the term "action figure". G.I. Joe's appeal to children has made it an American icon among toys.The G.I. Joe trademark has been used by Hasbro for several different toy lines, although only two have been successful. The original 12-inch (30 cm) line introduced on February 2, 1964 centered on realistic action figures. In the United Kingdom, this line was licensed to Palitoy and known as Action Man. In 1982 the line was relaunched in a 3.75-inch (9.5 cm) scale complete with vehicles, playsets, and a complex background story involving an ongoing struggle between the G.I. Joe Team and the evil Cobra Command which seeks to take over the Free World through terrorism. As the American line evolved into the Real American Hero series, Action Man also changed, by using the same molds and being renamed as Action Force. Although the members of the G.I. Joe team are not superheroes, they all had expertise in areas such as martial arts, weapons, and explosives.G.I. Joe was inducted into the National Toy Hall of Fame at The Strong in Rochester, New York, in 2003.

Gauge theory

In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations.

The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian. The transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called gauge invariance). When such a theory is quantized, the quanta of the gauge fields are called gauge bosons. If the symmetry group is non-commutative, then the gauge theory is referred to as non-abelian gauge theory, the usual example being the Yang–Mills theory.

Many powerful theories in physics are described by Lagrangians that are invariant under some symmetry transformation groups. When they are invariant under a transformation identically performed at every point in the spacetime in which the physical processes occur, they are said to have a global symmetry. Local symmetry, the cornerstone of gauge theories, is a stronger constraint. In fact, a global symmetry is just a local symmetry whose group's parameters are fixed in spacetime (the same way a constant value can be understood as a function of a certain parameter, the output of which is always the same).

Gauge theories are important as the successful field theories explaining the dynamics of elementary particles. Quantum electrodynamics is an abelian gauge theory with the symmetry group U(1) and has one gauge field, the electromagnetic four-potential, with the photon being the gauge boson. The Standard Model is a non-abelian gauge theory with the symmetry group U(1) × SU(2) × SU(3) and has a total of twelve gauge bosons: the photon, three weak bosons and eight gluons.

Gauge theories are also important in explaining gravitation in the theory of general relativity. Its case is somewhat unusual in that the gauge field is a tensor, the Lanczos tensor. Theories of quantum gravity, beginning with gauge gravitation theory, also postulate the existence of a gauge boson known as the graviton. Gauge symmetries can be viewed as analogues of the principle of general covariance of general relativity in which the coordinate system can be chosen freely under arbitrary diffeomorphisms of spacetime. Both gauge invariance and diffeomorphism invariance reflect a redundancy in the description of the system. An alternative theory of gravitation, gauge theory gravity, replaces the principle of general covariance with a true gauge principle with new gauge fields.

Historically, these ideas were first stated in the context of classical electromagnetism and later in general relativity. However, the modern importance of gauge symmetries appeared first in the relativistic quantum mechanics of electrons – quantum electrodynamics, elaborated on below. Today, gauge theories are useful in condensed matter, nuclear and high energy physics among other subfields.

George G. Cleveland

George Grant Cleveland (born May 9, 1939), is a Republican member of the North Carolina House of Representatives, representing the 14th District since 2004. He is a 25-year veteran of the United States Marine Corps.He currently resides in Jacksonville, North Carolina, where he has lived for over 25 years. In 2012, Cleveland generated controversy when he stated that the state of North Carolina has "no one in the state of North Carolina living in extreme poverty" during a debate in the House regarding preschool funding. In a conflicting statement, the non-profit group Action for Children in North Carolina cited statistics claiming one in ten North Carolina children live in extreme poverty.In February 2017, Cleveland joined with Representatives Michael Speciale (R-Craven), and Larry Pittman (R-Cabarrus) in proposing a constitutional amendment that would allow North Carolina voters to repeal Article I, Section 4 of the North Carolina Constitution. This article declares "This State shall ever remain a member of the American Union; the people thereof are part of the American nation," and prohibits the state from seceding from the United States of America, and its inclusion in North Carolina’s 1868 constitution was a condition for being readmitted into the Union after the Civil War.2017 Session

HB 34

During the 2017 session, Cleveland introduced a bill to budget the funds to purchase for the North Carolina State Highway Patrol three rescue helicopters.

2018 Session

H.B. 1050

A bill to authorize the Department of Military and Veterans Affairs to apply for Federal Funds for the Expansion of Sandhills 4 State Vets Cemetery and Western Carolina State Cemetery. 2018

In November 2108, Cleveland beat Isaiah Johnson by almost 20 points.

Georges Besse

Georges Besse (25 December 1927, in Clermont-Ferrand, France – 17 November 1986, in Paris) was a French businessman who led several large state-controlled French companies during his lifetime. He was assassinated outside his home by the terrorist group Action directe. At the time of his death he was the CEO of French car manufacturer Renault.

AREVA operates the Georges Besse II uranium enrichment plant.[2]

Group action (sociology)

In sociology, a group action is a situation in which a number of agents take action simultaneously in order to achieve a common goal; their actions are usually coordinated.

Group action will often take place when social agents realize they are more likely to achieve their goal when acting together rather than individually. Group action differs from group behaviours, which are uncoordinated, and also from mass actions, which are more limited in place.

Group action is more likely to occur when the individuals within the group feel a sense of unity with the group, even in personally costly actions.

Homogeneous space

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 (rigid geometry), diffeomorphism (differential geometry), or homeomorphism (topology). Some authors insist that the action of G be faithful (non-identity elements act non-trivially), 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.

Invariant (mathematics)

In mathematics, an invariant is a property, held by a class of mathematical objects, which remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class.

Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged, for example conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important step in the process of classifying mathematical objects.

Linear algebraic group

In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation MTM = 1 where MT is the transpose of M.

Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R, as can many noncompact groups such as the simple Lie group SL(n,R).) The simple Lie groups were classified by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include Maurer, Chevalley, and Kolchin (1948). In the 1950s, Armand Borel constructed much of the theory of algebraic groups as it exists today.

One of the first uses for the theory was to define the Chevalley groups.

Mass action (sociology)

Mass action in sociology refers to the situations where a large number of people behave simultaneously in a similar way but individually and without coordination.

For example, at any given moment, many thousands of people are shopping - without any coordination between themselves, they are nonetheless performing the same mass action. Another, more complicated example would be one based on a work of 19th-century German sociologist Max Weber, The Protestant Ethic and the Spirit of Capitalism: Weber wrote that capitalism evolved when the Protestant ethic influenced large number of people to create their own enterprises and engage in trade and gathering of wealth. In other words, the Protestant ethic was a force behind an unplanned and uncoordinated mass action that led to the development of capitalism.

A bank run is mass action with sweeping implications. Upon hearing news of a bank's anticipated insolvency, hundreds or thousands of bank depositors simultaneously rush down to a bank branch to withdraw their deposits, and protect their savings.More developed forms of mass actions are group behavior and group action.

Moment map

In mathematics, specifically in symplectic geometry, the momentum map (or moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums.

Permutation group

In mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself). The group of all permutations of a set M is the symmetric group of M, often written as Sym(M). The term permutation group thus means a subgroup of the symmetric group. If M = {1,2,...,n} then, Sym(M), the symmetric group on n letters is usually denoted by Sn.

By Cayley's theorem, every group is isomorphic to some permutation group.

The way in which the elements of a permutation group permute the elements of the set is called its group action. Group actions have applications in the study of symmetries, combinatorics and many other branches of mathematics, physics and chemistry.

Shunning

Shunning can be the act of social rejection, or emotional distance. In a religious context, shunning is a formal decision by a denomination or a congregation to cease interaction with an individual or a group, and follows a particular set of rules. It differs from, but may be associated with, excommunication.

Social rejection occurs when a person or group deliberately avoids association with, and habitually keeps away from an individual or group. This can be a formal decision by a group, or a less formal group action which will spread to all members of the group as a form of solidarity. It is a sanction against association, often associated with religious groups and other tightly knit organizations and communities. Targets of shunning can include persons who have been labeled as apostates, whistleblowers, dissidents, strikebreakers, or anyone the group perceives as a threat or source of conflict. Social rejection has been established to cause psychological damage and has been categorized as torture or punishment. Mental rejection is a more individual action, where a person subconsciously or willfully ignores an idea, or a set of information related to a particular viewpoint. Some groups are made up of people who shun the same ideas.Social rejection has been and is a punishment used by many customary legal systems. Such sanctions include the ostracism of ancient Athens and the still-used kasepekang in Balinese society.

Symmetry group

In group theory, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation. For a space with a metric, it is a subgroup of the isometry group of the space concerned. If not stated otherwise, this article considers symmetry groups in Euclidean geometry, but the concept may also be studied in more general contexts as expanded below.

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