Milne - Group Theory.. - Free

36 4 GROUPS ACTING ON SETS
The group G is said to act transitively on X if there is only one orbit, i.e., for any
two elements x and y of X, thereexistsag ∈ G suchthat gx = y.
For example, S n acts transitively on {1, 2, ...n}. For any subgroup H of a group
G, G acts transitively on G/H. ButG (almost) never acts transitively on G (or G/N
or N) by conjugation.
The group G acts doubly transitively on X if for any two pairs (x, x ′ ), (y, y ′ )of
elements of X, there exists a (single) g ∈ G suchthat gx = y and gx ′ = y ′ . Define
k-fold transitivity, k ≥ 3, similarly.
Stabilizers
The stabilizer (or isotropy group) ofanelementx ∈ X is
Stab(x) ={g ∈ G | gx = x}.
It is a subgroup, but it need not be a normal subgroup. In fact:
Lemma 4.4. If y = gx, thenStab(y) =g · Stab(x) · g −1 .
Proof. Certainly, if g ′ x = x, then
(gg ′ g −1 )y = gg ′ x = gx = y.
Hence Stab(y) ⊃ g · Stab(x) · g −1 .Conversely,ifg ′ y = y, then
(g −1 g ′ g)x = g −1 g ′ (y) =g −1 y = x,
and so g −1 g ′ g ∈ Stab(x), i.e., g ′ ∈ g · Stab(x) · g −1 .
Clearly
⋂
Stab(x) =Ker(G → Sym(X)),
which is a normal subgroup of G. If ⋂ Stab(x) ={1}, i.e., G↩→ Sym(X), then
G is said to act effectively. It acts freely if Stab(x) = 1 for all x ∈ X, i.e., if
gx = x ⇒ g =1.
Example 4.5. (a) Let G act on G by conjugation. Then
Stab(x) ={g ∈ G | gx = xg}.
This group is called the centralizer C G (x) ofx in G. It consists of all elements of G
that commute with, i.e., centralize, x. The intersection
⋂
CG (x) ={g ∈ G | gx = xg ∀x ∈ G}
is a normal subgroup of G, called the centre Z(G) ofG. It consists of the elements
of G that commute with every element of G.
(b) Let G act on G/H by left multiplication. Then Stab(H) = H, and th e
stabilizer of gH is gHg −1 .