Group Actions

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 4 August 2012

Group Actions

Let G be a group.

Examples of group actions are given below in this section and in the Exercises.

Let G be a group and let X be a G-set. Let xX.

g1x =g1g4x g3x =g3g4x x=g4x g4 g3 g2 g2x =g2g4x g1 S

Suppose G is a group acting on a set X and let xX and gG. Then

  1. (a) Gx is a subgroup of G.
  2. (b) Ggx =gGx g-1.


  1. To show:
    1. If h1,h2 Gx then h1h2 Gx.
    2. 1Gx.
    3. If hGx then h-1Gx.
    4. (Proof) Assume h1,h2 Gx. Then (h1h2)x =h1( h2x =h1x=x. So h1,h2 Gx.
    5. (Proof) Since 1x=x, 1Gx.
    6. (Proof) Assume hGx. Then h-1x =h-1 (hx) =(h-1 h)x =1x=x. So h-1 Gx.
    So Gs is a subgroup of G.
  2. To show:
    1. Ggx gGx g-1.
    2. gGx g-1 Ggx.
    3. (Proof) Assume hGgx.
    4. Then hgs=hx.
    5. So g-1 hgx=x
    6. So g-1 hgGx.
    7. Since h=g( g-1hg g-1 ,hgGx g-1 .
    8. So Ggx gGs g-1.
    9. (Proof) Assume hgGx g-1.
    10. So h= gag-1 for some aGx.
    11. Then hgx= (gag-1 ) gx=gax=gx .
    12. So hGgx gGx g-1.
    So Ggx =gGx g-1.

The following is an analogue of Proposition 1.1.3.

Let G be a group which acts on a set X. Then the orbits partition the set X.

To show:
  1. If xXthen xGy for some yX.
  2. If x1,x2 S and Gx1 Gx2 then Gx1 =Gx2.
  3. (Proof) Assume xX.
  4. Then, since x=1x, xGx.
  5. (Proof) Assume x1,x2 S and that Gx1 Gx2.
  6. Then let t Gx1 Gx2.
  7. So y= g1x1 and y= g2x2 for some elements g1,g2 G.
  8. So x1= g1-1 g2x2 and x2 =g2-1g1x1.
  9. To show: Gx1 =Gx2.
    1. To show:
    2. Gx1 Gx2.
    3. Gx2 Gx1.
    4. (Proof) Let y1 Gx1.
    5. So y1= h1x1 for some h1G.
    6. Then y1= h1x1 =h1 g1-1 g2x2Gx2. So Gx1 Gx2.
    7. (Proof) Let y2Gx2 .
    8. So y2= h2x2 for some h2G.
    9. Then y2 =h2x2 =h2 g2-1 g1x1 Gs1. So Gx2Gx1.
    So Gx1 =Gx2.
So the orbits partition S.

If G is a group acting on a set X and Gxi denote the orbits of the action of G on X then Card(X) =distinct orbits Card(Gxi) .


By Proposition 1.2, Xis a disjoint union of orbits.
So Card(X) is the sum of the cardinalities of the orbits.

It is possible to view the stabiliser Gs of an element sS as an analogue of the kernel of a homomorphism and the orbit Gs of an element sS as an analogue of the image of a homomorphism. One might say group actions   α:G×X X are to group homomorphisms f:GH as stabilisers Gx are to kernels kerf as orbits Gx are to images imf.

From this point of view the following corollary is an analogue of Corollary 1.1.5.

Let G be a group acting on a set X and let xX. If Gx is the orbit containing x and Gx is the stabiliser of x then |G: Gx| =Card(Gx) where |G: Gx| is the index of GxG.


Recall that |G:Gx| =Card(G/Gx) .
To show: There is a bijective map φ:G/Gx Gx. Define φ: G/ Gx Gx gGx gx. To show:

  1. φ is well defined.
  2. φ is bijective.
  3. (Proof) To show
    1. φ(gGx) Gx for every gG.
    2. If g1Gx =g2Gx then φ(g1 Gs) =φ( g2Gs) .
    3. (Proof) Is clear from the definition of φ that φ(gGx) =gxGx.
    4. (Proof) Assume g1,g2 G and g1Gx =g2Gx.
    5. Then g1 =g2h for some hGs.
    6. To show: g1s =g2s.
      1. Then g1x =g2hx =g2x, since hGx.
      So φ(g1Gx ) =φ(g2Gx ) .
    So φ is well defined.
  4. To show:
    1. φ is injective, i.e. if φ( g1Gx) = φ(g2G2 ) then g1Gx =g2Gx .
    2. φ is surjective, i.e. if gx Gx then there exists hGx G/Gx such that φ(hGs) =gs.
    3. (Proof) Assume φ(g1 Gs) =φ(g2Gs ).
    4. Then g1x =g2x.
    5. So s= g1-1 g2x and g2-1 g1x=x.
    6. So g1-1 g2Gx and g2-1 g1Gx.
    7. To show: φ is injective.
      1. To show: g1Gx =g2Gx.
        1. To show:
        2. baa) g1Gx g2Gx.
        3. bab) g2Gx g1Gx.
        4. baa) (Proof) Let k1 g1Gx.
        5. So k1 =g1h1 for some h1Gx .
        6. Then k1 =g1h1 =g1 g1-1 g2 g2-1 g1h1 =g2( g2-1 g1h1) g2Gx. So g1Gx g1Gx .
        7. bab) (Proof) Let k2 g2Gx.
        8. So k2 =g2h2 for some h2Gx .
        9. Then k2 =g2h2 =g2 g2-1 g1 g1-1 g2h2 =g2( g2-1 g1h1) g1Gx. So g2Gx g1Gx.
        So g1Gx =g2Gx.
      So φ is injective.
    8. To show: φ is surjective.
      1. Assume yGx.
      2. Then y=gx for some g G.
      3. Thus φ(gGx) =gx=y.
      So φ is surjective.
    9. So φ is bijective.

Let G be a group acting on a set X. Let Gx denote the stabiliser of x and let Gx denote the orbit of x. Then Card(G) =Card(Gx) Card(Gx).

Multiply both sides of the identity in Proposition 1.4 by Card(Gx) and use Corollary 2.3 from 'Groups, Basic Definitions and Cosets'.


Let X be a subset of a group G.

Let H be a subgroup of G and let NH be the normaliser of H in G. Then

  1. H is a normal subgroup of NH.
  2. If K is a subgroup of G such that HKG and H is a normal subgroup of K then KNH.

  1. Let kK.
  2. To show: kNH.
    1. khk-1 H for all hH.
    2. This is true since His normal in K.
  3. So KNH.
  4. This is the special case of b) when K=H.

This proposition says that NH is the largest subgroup of G such that H is normal in this subgroup.

Let G be a group and let 𝒮 be the set of subsets of G. Then

  1. G acts on 𝒮 by G×𝒮 𝒮 (gS) gSg-1 where gSg-1 ={gsg-1 | sS} . We say that G acts on 𝒮 by conjugation.
  2. If S is a subset of G then NS is the stabiliser of S under the action of G on 𝒮 by conjugation.


  1. To show:
    1. α is well defined.
    2. α(1S)=S, for all S𝒮.
    3. α(g α(hS)) =α(gh)S) , for all g,hG and S𝒮.
    4. (Proof) To show:
      1. aaa) gS g-1 𝒮.
      2. aab) If S=T and g=h then gSg-1 = hTh-1.
      Both of these are clear from the definitions.
    5. (Proof) Let S𝒮.
    6. Then α(1S) =1S 1-1 =S.
    7. (Proof) Let g,hG and S𝒮.
    8. Then α(g,α( h,S) = α(g, hSh-1) =g( hSh-1 ) g-1 = (gh)S( h-1 g-1 ) =(gh)S (gh)-1 =α(gh,S) .
  2. This follows immediately from the definitions of NS and of stabiliser.

Let G be a group. Then

  1. G acts on G by G×G G (gs) gsg-1. We say that G acts on itself by conjugation.
  2. Two elements g1,g2 G are conjugate if and only if they are in the same orbit under the action of G on itself by conjugation.
  3. The conjugacy class 𝒞g of gG is the orbit of g under the action of G on itself by conjugation.
  4. The centraliser Zg of gG is the stabiliser of gG under the action of G on itself by conjugation.

  1. The proof is exactly the same as in the proof of (a) in Proposition 2.2. One simply replaces all the capital S's by lower case s's.
  2. and (c) and (d) follow from the definitions.

Let S be a subset of a group G. The centraliser of S in G is the set ZS ={xG | xsx-1 =s for all sS}.

Let Gs be the stabiliser of sG under the action of G on itself by conjugation. Then

  1. For each subset SG, ZS= sS Gs.
  2. Z(G)= ZG, where Z(G) denotes the center of G.
  3. sZ(G) if and only if Zs=G.
  4. sZ(G) if and only if 𝒞s ={s}.

    1. Assume sZs.
    2. Then sxs-1 =s, for all sS.
    3. So xGs for all sS.
    4. So x s SGs.
    5. So Zs sS Gs.
    6. Assume x sS Gs.
    7. Then xs x-1 =s, for all sS .
    8. So xZs.
    9. So sS Gs.
  1. This is clear from the definitions of ZG and Z(G).
  2. : Let sZ(G).
    To show: ZS=G.
    1. By definition ZSG.
    2. To show: GZS.
      1. Let gG.
      2. Then gsg-1 =s since sZ(G).
      3. So gZS.
    3. So GZS.
    So ZS=G.
    Assume ZS=G.
    Then gsg-1 =s, for all gG.
    So sg=gs, for all gG.
    So sZ(G).
  3. : Assume sZ(G).
    Then gsg-1 =s, for all sG.
    So 𝒞s={ gsg-1 | gG} ={s}.
    : Assume 𝒞s ={s}.
    Then gsg-1 =s, for all gG.
    So sZ(g).

(The Class Equation) Let 𝒞gi denote the conjugacy classes in a group G and let |𝒞gi | denote Card( 𝒞gi). Then |G| =| Z(G)| +| Z(G)| >1 Card(𝒞gi) .

By Corollary 1.3 and the fact that 𝒞gi are the orbits of G acting on itself by conjugation we know that |G| = 𝒞gi Card(𝒞gi ) . By Lemma 2.4 (d) we know that Z(G) = |𝒞gi | =1 𝒞gi. So |G|= | 𝒞gi | =1 Card(𝒞gi ) + |𝒞gi | >1 Card(𝒞gi ) =Card ( Z(G) ) + | 𝒞gi | >1 Card(𝒞gi ) .

Notes and References

These notes are written to highlight the analogy between groups and group actions, rings and modules, and fields and vector spaces.


[Ram] A. Ram, Notes in abstract algebra, University of Wisconsin, Madison 1993-1994.

[Bou] N. Bourbaki, Algèbre, Chapitre 9: Formes sesquilinéaires et formes quadratiques, Actualités Sci. Ind. no. 1272 Hermann, Paris, 1959, 211 pp. MR0107661.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.

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