Group action: Difference between revisions

Revision as of 20:50, 8 December 2008

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
VIEW: Definitions built on this | Facts about this: (facts closely related to Group action, all facts related to Group action) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |[SHOW MORE]

View a complete list of basic definitions in group theory | Go through a guided tour for beginners to this wiki

We follow the convention of action on the left. Please refer Convention:Group action on left

Definition

Definition in terms of action

A group action $α$ of a group $G$ on a set $S$ is a map $α : G \times S \to S$ such that the following two conditions are satisfied:

$α (g h, s) = α (g, α (h, s))$
$α (e, s) = s$ (here, $e$ is the identity element of $G$ ).

The above is the definition of left action. For the definition of a right action of a group, refer right action of a group.

Definition in terms of homomorphisms

A group action on a set or an action of a group on a set is a group homomorphism from the group to the symmetric group on the set.

In symbols, a group action of a group $G$ on a set $S$ is a homomorphism $ρ : G \to S y m (S)$ where $S y m (S)$ denotes the symmetric group on $S$ .

Equivalence of definitions

Further information: Equivalence of definitions of group action

Convenience of notation

For convenience, we omit the symbols $α$ or $ρ$ , and write the action of $g$ on $s$ as $g . s$ , or sometimes just as $g s$ .

We can then rewrite the first condition as:

$g (h s) = (g h) s$

This is just like associativity, and hence we can drop the parenthesization, so we often write $g h s$ for either of the above.

Related notions

Right action of a group
Monoid action: This is the corresponding notion of action for a group without inverses

Also refer Category:Group action properties

Terminology

Orbit

Further information: orbit under group action

Suppose $G$ is a group acting on a set $S$ . Then, for any point $s \in S$ , the orbit of $s$ under the action of $G$ , denoted $G . s$ , is defined as:

${t \in S ∣ \exists g \in G, g . t = s}$

In other words, the orbit of a point is the st of all points that can be reached from that point under the action of the group.

Because of the reversibility of the action of elements of the group, it turns out that if $t$ is in the orbit of $s$ , $s$ is also in the orbit of $t$ . Specifically, if $g . s = t$ , then $g^{- 1} . t = s$ . Hence we can talk of the relation of being in the same orbit. This relation is reflexive (because of the identity element), symmetric (because of invertibility) and transitive (because of the homomorphism nature of the group action), and hence gives an equivalence relation. The equivalence relation thus partitions $S$ into a disjoint union of orbits.

Stabilizer

Further information: Point-stabilizer

Given a group $G$ acting on a set $S$ , the point-stabilizer of $s \in S$ , also termed the isotropy group or isotropy subgroup at $S$ , denoted $S t a b_{G} (s)$ , is defined as:

${h \in G ∣ h s = s}$

In other words, it is those elements of the group that fix $s$ .

Some true facts:

The orbit of $s$ can be naturally identified with the coset space of $G / H$ , where $H$ is the isotropy subgroup of $s$ . For full proof, refer: Fundamental theorem of group actions
If $g s = t$ , the isotropy subgroup at $s$ and $t$ are conjugate by $g$ ; in other words:

$g S t a b_{G} (s) g^{- 1} = S t a b_{G} (t)$

@@ Line 2: / Line 2: @@
 ''We follow the convention of action on the left. Please refer [[Convention:Group action on left]]''
+<section begin=beginner/>
 ==Definition==
+===Definition in terms of action===
-===Symbol-free definition===
+A '''group action''' <math>\alpha</math> of a group <math>G</math> on a set <math>S</math> is a map <math>\alpha: G \times S \to S</math> such that the following two conditions are satisfied:
-A '''group action''' on a set or an action of a group on a set is a [[group homomorphism]] from the group to the [[symmetric group]] on the set.
+* <math>\alpha(gh,s) = \alpha(g,\alpha(h,s))</math>
+* <math>\alpha(e,s) = s</math> (here, <math>e</math> is the identity element of <math>G</math>).
-===Definition with symbols===
+The above is the definition of left action. For the definition of a ''right'' action of a group, refer [[right action of a group]].
+<section end=beginner/>
+===Definition in terms of homomorphisms===
-A '''group action''' <math>\alpha</math> of a group <math>G</math> on a set <math>S</math> is either of the following equivalent things:
+A '''group action''' on a set or an action of a group on a set is a [[group homomorphism]] from the group to the [[symmetric group]] on the set.
-* <math>\alpha: G \times S \to S</math> such that <math>\alpha(gh,s) = \alpha(g,\alpha(h,s))</math> and <math>\alpha(e,s) = s</math>.
-* A homomorphism <math>\rho: G \to Sym(S)</math> where <math>Sym(S)</math> denotes the symmetric group on <math>S</math>.
-The above is the definition of left action. For the definition of a ''right'' action of a group, refer [[right action of a group]].
+In symbols, a group action of a group <math>G</math> on a set <math>S</math> is a homomorphism <math>\rho: G \to \operatorname{Sym}(S)</math> where <math>\operatorname{Sym}(S)</math> denotes the symmetric group on <math>S</math>.
 ===Equivalence of definitions===
 {{further|[[Equivalence of definitions of group action]]}}
+<section begin=beginner/>
-Let us see why the two definitions given above are equivalent.
-Suppose we are given a map <math>\alpha:G \times S \to S</math> satisfying the conditions:
-<math>alpha(gh,s) = \alpha(g,\alpha(h,s))</math>
-and:
-<math>\alpha(e,s) = s</math>
-We now construct the map <math>\rho</math>. For every <math>g \in G</math>, <math>\rho(g)</math> is defined as the map:
-<math>s \mapsto \alpha(g,s)</math>
-Clearly, <math>\rho(g)</math> is a function from <math>S</math> to <math>S</math>, and <math>\rho(e)</math> is the identity map (by the second assumption on <math>\alpha</math>. The fact that <math>\rho(g)</math> is in <math>Sym(S)</math> for every <math>g</math> holds because:
-<math>\alpha(e,s) = \alpha(g^{-1}g,s) = \alpha(g^{-1}\alpha(g,s)) = \rho(g^{-1})(\rho(g)s)</math>
-Hence <math>\rho(g^{-1})</matH> is a left inverse for <matH>\rho(g)</math>, and a similar argument shows that it is a right inverse. Thus, <math>\rho(g)</matH> is an invertible map from <math>S</math> to <math>S</math>, hence an element of <math>Sym(S)</math>.
-Finally the fact that <math>\rho</math> is a homomorphism follows from the first condition.
-The map the other way around is similar. Given a homomorphism <math>\rho:G \to Sym(S)</math>, the map <math>\alpha</math> is given by:
-<math>\alpha(g,s) = \rho(g)s</math>
-Again, <math>\rho</math> being a homomorphism guarantees that <math>\alpha</math> satisfies the required properties.
 ===Convenience of notation===
@@ Line 58: / Line 32: @@
 This is just like [[associativity]], and hence we can drop the parenthesization, so we often write <math>ghs</math> for either of the above.
+<section end=beginner/>
 ==Related notions==
@@ Line 65: / Line 39: @@
 Also refer [[:Category:Group action properties]]
+<section begin=beginner/>
 ==Terminology==
@@ Line 78: / Line 52: @@
 In other words, the orbit of a point is the st of all points that can be ''reached'' from that point under the action of the group.
-Because of the ''reversibility'' of the action of elements of the group, it turns out that if <math>t</math> is in the orbit of <math>s</math>, <math>s</math> is also in the orbit of <math>t</math>, and hence we can talk of the relation of being ''in the same orbit''. This relation is reflexive (because of the identity element), symmetric (because of invertibility) and transitive (because of the homomorphism nature of the group action), and hence gives an equivalence relation. The equivalence relation thus partitions <math>S</math> into a disjoint union of orbits.
+Because of the ''reversibility'' of the action of elements of the group, it turns out that if <math>t</math> is in the orbit of <math>s</math>, <math>s</math> is also in the orbit of <math>t</math>. Specifically, if <math>g.s = t</math>, then <math>g^{-1}.t = s</math>. Hence we can talk of the relation of being ''in the same orbit''. This relation is reflexive (because of the identity element), symmetric (because of invertibility) and transitive (because of the homomorphism nature of the group action), and hence gives an equivalence relation. The equivalence relation thus partitions <math>S</math> into a disjoint union of orbits.
 ===Stabilizer===
@@ Line 88: / Line 62: @@
 <math>\{ h \in G \mid \ hs = s \}</math>
-In other words, it is those elements of the group that ''fix'' <math>s</math>. Some true facts:
+In other words, it is those elements of the group that ''fix'' <math>s</math>.
+<section end=beginner/>
+Some true facts:
-* The orbit of <math>s</math> can be naturally identified with the [[coset space]] of <math>G/H</math>, where <math>H</math> is the isotropy subgroup of <math>s</math>. {{fillin}}
+* The orbit of <math>s</math> can be naturally identified with the [[coset space]] of <math>G/H</math>, where <math>H</math> is the isotropy subgroup of <math>s</math>. {{proofat|[[Fundamental theorem of group actions]]}}
 * If <math>gs = t</math>, the isotropy subgroup at <math>s</math> and <math>t</math> are conjugate by <math>g</math>; in other words:
 <math>g Stab_G(s) g^{-1} = Stab_G(t)</math>