group actions and homomorphisms
Notes on group actions and homomorphisms
Let G be a group, X a non-empty set and SX the symmetric group
of X,
i.e. the group of all bijective
maps on X. ⋅ may denote a left
group
action
of G on X.
-
1.
For each g∈G and x∈X we define
fg:X→X,x↦g⋅x. Since fg-1(fg(x))=g-1⋅(g⋅x)=x for each x∈X, fg-1 is the inverse
of fg. so fg is bijective and thus element of SX. We define F:G→SX,F(g)=fg for all g∈G. This mapping is a group homomorphism
: Let g,h∈G,x∈X. Then
F(gh)(x) =fgh(x)=(gh)⋅x=g⋅(h⋅x) =(fg∘fh)(x)=(F(g)∘F(h))(x) for all x∈X implies F(gh)=F(g)∘F(h). — The same is obviously true for a right group action.
-
2.
Now let F:G→Sx be a group homomorphism, and let f:G×X→X,(g,x)↦F(g)(x) satisfy
-
(a)
f(1G,x)=F(1g)(x)=x for all x∈X and
-
(b)
f(gh,x)=F(gh)(x)=(F(g)∘F(h)(x)=F(g)(F(h)(x))=f(g,f(h,x)),
so f is a group action induced by F.
-
(a)
Characterization of group actions
Let G be a group acting on a set X. Using the same notation as above, we have for each g∈ker(F)
F(g)=idx=fg⇔g⋅x=x,∀x∈X⇔g∈∪x∈XGx | (1) |
and it follows
ker(F)=⋂x∈XGx. |
Let G act transitively on X. Then for any x∈X, X is the
orbit G(x)
of x. As shown in “conjugate stabilizer subgroups’, all stabilizer
subgroups
of elements y∈G(x) are conjugate subgroups
to Gx in G. From
the above it follows that
ker(F)=⋂g∈GgGxg-1. |
For a faithful operation of G the condition g⋅x=x,∀x∈X→g=1G is equivalent
to
ker(F)={1G} |
and therefore F:G→SX is a monomorphism.
For the trivial operation of G on X given by g⋅x=x,∀g∈G the stabilizer subgroup Gx is G for all x∈X, and thus
ker(F)=G. |
If the operation of G on X is free, then Gx={1G},∀x∈X, 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
π⋅i:= |
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 | Derivation |
Classification | msc 20A05 |
Related topic | GroupHomomorphism |