Milne - Group Theory.. - Free

34 4 GROUPS ACTING ON SETS
4 Groups Acting on Sets
General definitions and results
Definition 4.1. Let X be a set and let G be a group. A left action of G on X is a
mapping (g, x) ↦→ gx: G × X → X suchthat
(a) 1x = x, for all x ∈ X;
(b) (g 1 g 2 )x = g 1 (g 2 x), all g 1 , g 2 ∈ G, x ∈ X.
The axioms imply that, for each g ∈ G, left translation by g,
g L : X → X,
x ↦→ gx,
has (g −1 ) L as an inverse, and therefore g L is a bijection, i.e., g L ∈ Sym(X). Axiom
(b) now says that
g ↦→ g L : G → Sym(X)
is a homomorphism. Thus, from a left action of G on X, we obtain a homomorphism
G → Sym(G), and, conversely, every such homomorphism defines an action of G on
X.
Example 4.2. (a) The symmetric group S n acts on {1, 2, ..., n}. Every subgroup H
of S n acts on {1, 2,...,n}.
(b) Every subgroup H of a group G acts on G by left translation,
H × G → G,
(h, x) ↦→ hx.
(c) Let H be a subgroup of G. If C is a left coset of H in G, then so also is gC
for any g ∈ G. In this way, we get an action of G on the set of left cosets:
G × G/H → G/H,
(g, C) ↦→ gC.
(d) Every group G acts on itself by conjugation:
G × G → G, (g, x) ↦→ g x = df gxg −1 .
For any normal subgroup N, G acts on N and G/N by conjugation.
(e) For any group G, Aut(G) actsonG.
A right action X × G → G is defined similarly. To turn a right action into a left
action, set g ∗ x = xg −1 . For example, there is a natural right action of G on the set
of right cosets of a subgroup H in G, namely,(C, g) ↦→ Cg, which can be turned into
a left action (g, C) ↦→ Cg −1 .
A morphism of G-sets (better G-map; G-equivariant map) isamapϕ: X → Y
suchthat
ϕ(gx) =gϕ(x), all g ∈ G, x ∈ X.
An isomorphism of G-sets is a bijective G-map; its inverse is then also a G-map.