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In "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 Transformation s of the set). A '''permutation representation''' of a group ''G'' is a representation of ''G'' as a group of Permutation s 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 finite-dimensional 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 imes X o X\,
denoted
:(g,x)\mapsto g\cdot x\,
which satisfies the following two axioms:

# (''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) '''''G''-set'''. 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 ''S''''X''.

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''·(''g·h'') = (''x''·''g'')·''h''
# ''x''·''e'' = ''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''. From a right action a left action can be constructed by composing with the inverse operation on the group. If ''r'' is a right action, then
:l : G imes M o M : (g, m) \mapsto r(m, g^{-1})
is a left action, since
:l(gh, m) = r(m, (gh)^{-1}) = m\cdot (h^{-1}g^{-1})
:: = (m\cdot h^{-1}) \cdot g^{-1} = l(h, m) \cdot g^{-1} = l(g, l(h, m))\,
and
:l(e, m) = r(m, e^{-1}) = m \cdot e = m .
Similarly, any left action can be converted into a right action. Therefore in the sequel we consider only left group actions, since right actions add nothing new.


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 right-action variant of the latter case: ''x''''g'' = ''g''−1''xg''. The latter action is often called the Conjugation action.

  • The Symmetric Group S''n'' and its Subgroup s 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.

  • 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 Group s which act on R''n''.

  • 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 Number s (R, +) acts on the Phase Space of " Well-behaved " 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 .

  • 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.

  • ''n''-transitive if for any pairwise distinct ''x''1, ..., ''x''''n'' and pairwise distinct ''y''1, ..., ''y''''n'' there is a ''g'' in ''G'' such that ''g''.''x''''k'' = ''y''''k'' for 1 ≤ ''k'' ≤ ''n''.

  • --- sharply n-transitive 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 G-torsor.


Every free action on a Non-empty 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'' = {''g'' in ''G'' : ''g''·''x'' = ''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'' = ''g''·''x''. The original action of ''G'' on ''X'' is faithful if and only if ''N'' = {''e''}.


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\{ g\cdot x \mid g \in G ight\}.

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 Class es 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 { ''g''·''y'' : ''y'' \in ''Y'' and ''g'' \in ''G''}. 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 = \{g \in G \mid g\cdot x = 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 ''G'' → Sym(''X'') is given by the Intersection of the stabilizers ''G''''x'' for all ''x'' in ''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 Coset s of ''Gx''. The standard quotient theorem of set theory then gives a natural Bijection between ''G''/''G''''x'' and ''Gx''. Specifically, the bijection is given by ''hGx'' \mapsto ''h''·''x''. This result is known as the orbit-stabilizer theorem.

If ''G'' and ''X'' are finite then the orbit-stabilizer theorem, together with Lagrange's Theorem , gives