Dictionary Definition
effectively adv
1 in an effective manner; "these are real
problems that can be dealt with most effectively by rational
discussion" [syn: efficaciously] [ant:
inefficaciously,
inefficaciously]
2 in actuality or reality or fact; "she is
effectively his wife"; "in effect, they had no choice" [syn:
in
effect]
User Contributed Dictionary
English
Adverb
 In an efficient or effective manner; with powerful effect.
 Essentially; for all practical purposes.
Translations
in an efficient or effective manner; with
powerful effect
 Finnish: tehokkaasti, vaikuttavasti
 French: efficacement
 Hungarian: hatékonyan, hathatósan, eredményesen
essentially
 Hungarian: tényleg, ténylegesen, valóban
 ttbc Portuguese: efetivamente °
Extensive Definition
In mathematics, a symmetry
group describes all symmetries of objects. This is
formalized by the notion of a group action: every element of the
group
"acts" like a bijective map (or "symmetry")
on some set. In this case, the group is also called a permutation
group (especially if the set is finite or not a vector space)
or transformation group (especially if the set is a vector space
and the group acts like linear
transformations of the set). A permutation representation of a
group G is a representation of G as a group of permutations of the set
(usually if the set is finite), and may be described as a group
representation of G by permutation
matrices, and is usually considered in the finitedimensional
case—it is the same as a group action of G on an ordered
basis of a vector space.
Definition
If G is a group
and X is a set, then a
(left) group action of G on X is a binary
function
 G \times X \to X\,
 (g,x)\mapsto g\cdot x\,
 (gh)·x = g·(h·x) for all g, h in G and x in X
 e·x = x for every x in X (where e denotes the identity element of G)
The set X is called a (left) Gset. The group G
is said to act on X (on the left).
From these two axioms, it follows that for every
g in G, the function which maps x in X to g·x is a bijective
map from X to X. Therefore, one may alternatively define a
group action of G on X as a group
homomorphism from G into the symmetric
group SX.
In complete analogy, one can define a right group
action of G on X as a function X × G → X by the
two axioms:
 x·(gh) = (x·g)·h
 x·e = x
 l : G \times M \to M : (g, m) \mapsto r(m, g^)
 l(gh, m) = r(m, (gh)^) = m\cdot (h^g^)

 = (m\cdot h^) \cdot g^ = l(h, m) \cdot g^ = l(g, l(h, m))\,
 l(e, m) = r(m, e^) = m \cdot e = m .
Examples
 The trivial action for any group G is defined by g·x=x for all g in G and all x in X; that is, the whole group G induces the identity permutation on X.
 Every group G acts on G in two natural but essentially different ways: g·x = gx for all x in G, or g·x = gxg−1 for all x in G. An exponential notation is commonly used for the rightaction variant of the latter case: xg = g−1xg. The latter action is often called the conjugation action.
 The symmetric group Sn and its subgroups act on the set by permuting its elements
 The symmetry group of a polyhedron acts on the set of vertices of that 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,R), special linear group SL(n,R), orthogonal group O(n,R), and special orthogonal group SO(n,R) are Lie groups which act on Rn.
 The Galois group of a field extension E/F acts on the bigger field E. So does every subgroup of the Galois group.
 The additive group of the real numbers (R, +) acts on the phase space of "wellbehaved" systems in classical mechanics (and in more general dynamical systems): 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 with (g·f)(x) equal to e.g. f(x + g), f(x) + g, f(x e^g), f(x) e^g, f(x+g) e^g, or f(x e^g)+g, but not f(x e^g+g)
 The quaternions with modulus 1, as a multiplicative group, act on R3: for any such quaternion z = \cos\frac + \sin\frac\,\hat\mathbf, the mapping f(x) = z x z* is a counterclockwise rotation through an angle \alpha\, about an axis v; −z is the same rotation; see quaternions and spatial rotation.
 The isometries of the plane act on the set of 2D images and patterns, such as a wallpaper pattern. 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.
 More generally, a group of bijections g: V → V acts on the set of functions x: V → W by (gx)(v) = x(g−1(v)) (or a restricted set of such functions that is closed under the group action). Thus a group of bijections of space induces a group action on "objects" in it.
Types of actions
The action of G on X is called
 transitive if for any two x, y in X there exists a g in G such
that g·x = y.
 sharply transitive if that g is unique; it is equivalent to regularity defined below.
 ntransitive if for any pairwise distinct x1, ..., xn and
pairwise distinct y1, ..., yn there is a g in G such that g.xk = yk
for 1 ≤ k ≤ n. A 2transitive action is also called
doubly transitive, a 3transitive action is also called triply
transitive, and so on.
 sharply ntransitive if there is exactly one such g.
 faithful (or effective) if for any two distinct g, h in G there exists an x in X such that g·x ≠ h·x; or equivalently, if for any g≠ e in G there exists an x in X such that g·x ≠ x.
 free or semiregular if for any two distinct g, h in G and all x in X we have g·x ≠ h·x; or equivalently, if g·x = x for some x implies g = e.
 regular (or simply transitive) if it is both transitive and free; this is equivalent to saying that for any two x, y in X there exists precisely one g in G such that g·x = y. In this case, X is known as a principal homogeneous space for G or as a Gtorsor.
 locally free if G is a topological group, and there is a neighbourood U of e in G such that the restriction of the action to U is free; that is, if g·x = x for some x and some g in U then g = e.
Every free action on a nonempty set is
faithful. A group G acts faithfully on X if and
only if the homomorphism G → Sym(X) has a trivial kernel.
Thus, for a faithful action, G is isomorphic to a permutation
group on X; specifically, G is isomorphic to its image in
Sym(X).
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) — a result known as Cayley's
theorem.
If G does not act faithfully on X, one can easily
modify the group to obtain a faithful action. If we define N = ,
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 = g·x. The original
action of G on X is faithful if and only if N = .
Orbits and stabilizers
Consider a group G acting on a set X. The orbit
of a point x in X is the set of elements of X to which x can be
moved by the elements of G. The orbit of x is denoted by Gx:
 Gx = \left\.
The defining properties of a group guarantee that
the set of orbits of X under the action of G form a partition
of X. The associated equivalence
relation is defined by saying x ~ y if and
only if there exists a g in G with g·x = 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 set of all orbits of X under the action of G
is written as X/G, and is called the quotient of the action; in
geometric situations it may be called the orbit space.
If Y is a subset of X, we write GY for the
set . We call the subset Y invariant under G if GY = Y (which is
equivalent to GY ⊆ Y). In that case, G also operates on Y. The
subset Y is called fixed under G if g·y = y for all g in G and all
y in Y. Every subset that's 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.
For every x in X, we define the stabilizer
subgroup of x (also called the isotropy group or little group) as
the set of all elements in G that fix x:
 G_x = \.
Orbits and stabilizers are not unrelated. For a
fixed x in X, consider the map from G to X given by g \mapsto g·x.
The image
of this map is the orbit of x and the coimage is the set of all left
cosets of Gx. The standard
quotient theorem of set theory then gives a natural bijection between G/Gx and Gx.
Specifically, the bijection is given by hGx \mapsto h·x. This
result is known as the orbitstabilizer theorem.
If G and X'' are finite then the orbitstabilizer
theorem, together with
Lagrange's theorem, gives
 Gx = [G\,:\,G_x] = G / G_x.
Note that if two elements x and y belong to the
same orbit, then their stabilizer subgroups, Gx and Gy, are
isomorphic
(or conjugate).
More precisely: if y = g·x, then Gy = gGx g−1. Points
with conjugate stabilizer subgroups are said to have the same
orbittype.
A result closely related to the orbitstabilizer
theorem is Burnside's
lemma:
 \leftX/G\right=\frac\sum_\leftX^g\right
The set of formal differences of finite Gsets
forms a ring
called the Burnside
ring, where addition corresponds to disjoint
union, and multiplication to Cartesian
product.
Group actions and groupoids
The notion of group action can be put in a
broader context by using the associated `action groupoid' G' = G\ltimes X
associated to the group action, thus allowing techniques from
groupoid theory such as presentations and fibrations. Further the
stabilisers 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 p: G'
\rightarrow G which is a `covering morphism of groupoids'. This
allows a relation between such morphisms and covering
maps in topology.
Morphisms and isomorphisms between Gsets
If X and Y are two Gsets, we define a morphism
from X to Y to be a function f : X → Y such that f(g.x) = g.f(x)
for all g in G and all x in X. Morphisms of Gsets are also called
equivariant
maps or Gmaps.
If such a function f is bijective, then its inverse is
also a morphism, and we call f an isomorphism and the two
Gsets 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 by left multiplication on the first coordinate.
 Every transitive G action is isomorphic to left multiplication by G on the set of left cosets of some subgroup H of G.
With this notion of morphism, the collection of
all Gsets forms a category;
this category is a topos.
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 × X → X is
continuous with respect to the product
topology of G × X. The space X is
also called a Gspace 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 Gspaces 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 G is a discrete group acting on a topological
space X, the action is properly
discontinuous if for any point x in X there is an open
neighborhood U of x in X, such that the set of all g \in G for
which g(U) \cap U \ne \emptyset consists of the identity only. 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 pathconnected
topological space X arises in this manner: the quotient map X
\mapsto X/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 generalised in the book Topology and
Groupoids referenced below to obtain the fundamental
groupoid of the orbit space of a discontinuous action of
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 a
symmetric square.
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×X → X×X that sends (g,x)\mapsto(gx,x)
is a proper
map.
Strongly continuous group action and smooth vector
If \alpha:V\times A\to A is an action of a
topological vector space V on an another topological vector space
A, one says that it is strongly continuous if for all a\in A, the
map v\mapsto\alpha_v(a) is continuous with respect to the
respective topologies.
Such an action induce an action on the space of
continuous function on A by (\alpha_vf)(x)=f(\alpha_v^x).
The space of smooth vector for the action \alpha
is the subspace of A of elements a such that x\mapsto\alpha_x(a) is
smooth, i.e. it is continuous and all derivatives are
continuous.
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.
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 group
action is then nothing but a functor from G to the category
of sets, and a group representation is a functor from G to the
category of vector spaces. In analogy, an action of a groupoid is a functor from the
groupoid to the category of sets or to some other category.
Without using the language of categories, one can
extend the notion of a group action on a set X by studying as well
its induced action on the power set of X.
This is useful, for instance, in studying the action of the large
Mathieu
group on a 24set and in studying symmetry in certain models of
finite
geometries.
See also
 Group with operators
 Act, the action of a monoid on a set.
 Gain graph
References
 Finite Group Theoryisbn=9780521786751}}
 Brown, Ronald (2006). Topology and groupoids, Booksurge PLC, ISBN 1419627228.
 [http://138.73.27.39/tac/reprints/articles/7/tr7abs.html 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.
effectively in German: Gruppenoperation
effectively in Esperanto: Grupa ago
effectively in Spanish: Acción
(matemática)
effectively in French: Action de groupe
(mathématiques)
effectively in Korean: 군의 작용
effectively in Italian: Azione di gruppo
effectively in Hebrew: פעולת חבורה
effectively in Polish: Działanie grupy na
zbiorze
effectively in Portuguese: Acção
(matemática)
effectively in Russian: Действие группы
effectively in Ukrainian: Дія групи
effectively in Contenese: 作用 (代數)
effectively in Chinese: 群作用
Synonyms, Antonyms and Related Words
ably,
adequately, advantageously, ardently, capably, cogently, competently, conveniently, dynamically, effectually, efficiently, eloquently, energetically, expressively, fervently, fluently, forcefully, forcibly, functionally, glibly, glowingly, graphically, handily, impressively, in glowing
terms, meaningfully, mightily, passionately, potently, powerfully, practically, productively, profitably, serviceably, smoothly, spiritedly, strikingly, tellingly, to advantage, to
good account, to good effect, to good purpose, to good use, to
profit, usefully,
vehemently, vigorously, vividly, warmly, well, with a vengeance, with
telling effect