group actions and homomorphisms

Let G be a group, X a non-empty set and SX the symmetric groupPlanetmathPlanetmath of X, i.e. the group of all bijectiveMathworldPlanetmathPlanetmath maps on X. may denote a left group actionMathworldPlanetmath of G on X.

  1. 1.

    For each gG and xX we define


    Since fg-1(fg(x))=g-1(gx)=x for each xX, fg-1 is the inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of fg. so fg is bijective and thus element of SX. We define F:GSX,F(g)=fg for all gG. This mapping is a group homomorphismMathworldPlanetmath: Let g,hG,xX. Then

    F(gh)(x) =fgh(x)=(gh)x=g(hx)

    for all xX implies F(gh)=F(g)F(h). — The same is obviously true for a right group action.

  2. 2.

    Now let F:GSx be a group homomorphism, and let f:G×XX,(g,x)F(g)(x) satisfy

    1. (a)

      f(1G,x)=F(1g)(x)=x for all xX and

    2. (b)


    so f is a group action induced by F.

Characterization of group actions

Let G be a group acting on a set X. Using the same notation as above, we have for each gker(F)

F(g)=idx=fggx=x,xXgxXGx (1)

and it follows


Let G act transitively on X. Then for any xX, X is the orbit G(x) of x. As shown in “conjugate stabilizer subgroups’, all stabilizerMathworldPlanetmath subgroupsMathworldPlanetmathPlanetmath of elements yG(x) are conjugate subgroupsMathworldPlanetmath to Gx in G. From the above it follows that


For a faithful operationMathworldPlanetmath of G the condition gx=x,xXg=1G is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to


and therefore F:GSX is a monomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

For the trivial operation of G on X given by gx=x,gG the stabilizer subgroup Gx is G for all xX, and thus


If the operation of G on X is free, then Gx={1G},xX, thus the kernel of F is {1G}–like for a faithful operation. But:

Let X={1,,n} and G=Sn. Then the operation of G on X given by


is faithful but not free.

Title group actions and homomorphisms
Canonical name GroupActionsAndHomomorphisms
Date of creation 2013-03-22 13:18:48
Last modified on 2013-03-22 13:18:48
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 15
Author CWoo (3771)
Entry type DerivationPlanetmathPlanetmath
Classification msc 20A05
Related topic GroupHomomorphism