Page 81 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
P. 81
mož Moravec: Some Topics in the Theory of Finite Groups 69
3.2.4 Group actions and Sylow’s theorems
Sylow theorems are central in the theory of finite groups, as they describe the structure
of such groups in terms of their subgroups of prime power order. These theorems are
closely related to another fundamental notion of group theory, actions.
Actions
An action of a group G on a non-empty set X is a map µ: X ×G → X satisfying the follow-
ing rules:
µ(µ(x , g ), h) = µ(x , g h),
µ(x , 1) = x
for all x ∈ X and g , h ∈ G . We usually suppres µ and write µ(x , g ) as x g . It is clear that the
above definition is equivalent to the fact that the map G → Sym X given by g → (x → x g )
is a homomorphism of groups. An action µ is faithful if the condition that µ(x , g ) =
µ(x , h) for all x ∈ X implies g = h.
Let G act on X . The relation ≡ defined on X by x ≡ y ⇔ ∃g ∈ G : x g = y is an
equivalence relation on X . The equivalence class of x ∈ X is called the orbit of x , and is
denoted by orbG (x ). The set of orbits of G on X will be denoted by X /G . The action is
said to be transitive if it has only one orbit, i.e., |X /G | = 1. For x ∈ X , the stabilizer of x is
stabG (x ) = {g ∈ G | x g = x }.
It is easy to see that stabG (x ) is a subgroup of G .
Example 3.2.14 A group G acts on itself by right multiplication, i.e., we have an action
G × G → G given by (g , h) → g · h = g h. It is not hard to see that this action is transitive
and faithful.
gap> G := Group((1,2,3),(2,3,4));;
gap> el := Elements( G);;
gap> OnRight(el[2], el[3]) = el[2] * el[3];
true
gap> orbit := Orbit(G, el[7], OnRight);
[ (1,3,2), (), (1,4,2), (1,2,3), (2,3,4), (1,4,3), (1,2)(3,4),
(1,3)(2,4), (2,4,3), (1,4)(2,3), (1,3,4), (1,2,4) ]
gap> Size( orbit ) = Order( G );
true
Example 3.2.15 A group G acts on itself by conjugation, i.e., (g , h) → g h . The orbits of this
actions are called the conjugacy classes of G . The stabilizer of g ∈ G is denoted by CG (g )
and called the centralizer of g in G .
3.2.4 Group actions and Sylow’s theorems
Sylow theorems are central in the theory of finite groups, as they describe the structure
of such groups in terms of their subgroups of prime power order. These theorems are
closely related to another fundamental notion of group theory, actions.
Actions
An action of a group G on a non-empty set X is a map µ: X ×G → X satisfying the follow-
ing rules:
µ(µ(x , g ), h) = µ(x , g h),
µ(x , 1) = x
for all x ∈ X and g , h ∈ G . We usually suppres µ and write µ(x , g ) as x g . It is clear that the
above definition is equivalent to the fact that the map G → Sym X given by g → (x → x g )
is a homomorphism of groups. An action µ is faithful if the condition that µ(x , g ) =
µ(x , h) for all x ∈ X implies g = h.
Let G act on X . The relation ≡ defined on X by x ≡ y ⇔ ∃g ∈ G : x g = y is an
equivalence relation on X . The equivalence class of x ∈ X is called the orbit of x , and is
denoted by orbG (x ). The set of orbits of G on X will be denoted by X /G . The action is
said to be transitive if it has only one orbit, i.e., |X /G | = 1. For x ∈ X , the stabilizer of x is
stabG (x ) = {g ∈ G | x g = x }.
It is easy to see that stabG (x ) is a subgroup of G .
Example 3.2.14 A group G acts on itself by right multiplication, i.e., we have an action
G × G → G given by (g , h) → g · h = g h. It is not hard to see that this action is transitive
and faithful.
gap> G := Group((1,2,3),(2,3,4));;
gap> el := Elements( G);;
gap> OnRight(el[2], el[3]) = el[2] * el[3];
true
gap> orbit := Orbit(G, el[7], OnRight);
[ (1,3,2), (), (1,4,2), (1,2,3), (2,3,4), (1,4,3), (1,2)(3,4),
(1,3)(2,4), (2,4,3), (1,4)(2,3), (1,3,4), (1,2,4) ]
gap> Size( orbit ) = Order( G );
true
Example 3.2.15 A group G acts on itself by conjugation, i.e., (g , h) → g h . The orbits of this
actions are called the conjugacy classes of G . The stabilizer of g ∈ G is denoted by CG (g )
and called the centralizer of g in G .
