MANTENIMIENTO DE SOFTWARE EN EL EUROFIGHTER (C.16)

kernel of the action. An element g belongs to the kernel if and only if gxg−1 = x for all x ∈ G, i.e., gx = xg or g belongs to Z(G), the center of G. It follows that Z(G) is the kernel of the conjugation representation.

A group G can also act on its set of subgroups by conjugation; thus if H ≤ G, define

g· H = gHg−1= {ghg−1 | h ∈ H }.

In this case the kernel consists of all group elements g such that gH g−1 = H for all

H ≤ G. This normal subgroup is called the norm of G; clearly it contains the center Z(G).

Exercises (5.1)

1. Complete the proof of (5.1.1).

2. Let (x, g)→ x·g be a right action of a group G on a set X. Define ρ : G → Sym(X) by ρ(g)(x)= x · g−1. Prove that ρ is a permutation representation of G on X. Why is the inverse necessary here?

3. Establish a bijection between the set of right actions of a group G on a set X and the set of permutation representations of G on X.

4. A right action of a group G on its underlying set is defined by x· g = xg. Verify that this is an action and describe the corresponding permutation representation of G, (this is called the right regular representation of G).

5. Prove that a permutation representation of a simple group is either faithful or trivial. 6. The left regular representation of a finite group is surjective if and only if the group has order 1 or 2.

7. Define a “natural” right action of a group G on the set of right cosets of a subgroup

H, and then identify the kernel of the associated representation.

8. Show that the number of (isomorphism types of) groups of order n is at most

(n!)[log2n] _{[Hint: a group of order n can be generated by}[log

2n] elements by Exer-

cise (4.1.10). Now apply Cayley’s Theorem.]

5.2

Orbits and stabilizers

We now proceed to develop the theory of group actions, introducing the fundamental concepts of orbit and stabilizer.

Let G be a group and X a non-empty set, and suppose that a left action of G on X is given. Then a binary relation∼

Gon X is defined by the rule:

a∼

for some g ∈ G. A simple verification shows that ∼

Gis an equivalence relation on the set X. The∼

G-equivalence class containing a is evidently

G· a = {g · a | g ∈ G},

which is called the G-orbit of a. Thus X is the union of all the distinct G-orbits and distinct G-orbits are disjoint, statements which follow from general facts about equivalence relations – see (1.2.2).

If X is the only G-orbit, the action of G on X – and the corresponding permutation representation of G – is called transitive. Thus the action of G is transitive if for each pair of elements a, b of X, there exists a g in G such that g· a = b. For example, the left regular representation is transitive and so is the left action of a group on the left cosets of a given subgroup.

Another important concept is that of a stabilizer. The stabilizer in G of an element

a∈ X is defined to be

StG(x)= {g ∈ G | g · x = x}.

It is easy to verify that St_G(a)is a subgroup of G. If St_G(a)= 1 for all a ∈ X, the

action is called semiregular. An action which is both transitive and semiregular is termed regular.

We illustrate these concepts by examining the group actions introduced in 5.1.

Example (5.2.1) Let G be any group.

(i) The left regular action of G is regular. Indeed (yx−1)x= y for any x, y ∈ G,

so it is transitive, while gx = x implies that g = 1, and regularity follows.

(ii) In the conjugation action of G on G the stabilizer of x in G consists of all g in G such that gxg−1 = x, i.e., gx = xg. This subgroup is called the centralizer of

xin G, the notation being

CG(x)= {g ∈ G | gx = xg}.

(iii) In the conjugation action of G on its underlying set the G-orbit of x is {gxg−1 _{| g ∈ G}, i.e., the set of all conjugates of x in G. This is called the con-}

jugacy class of x. The number of conjugacy classes of G is known as the class number.

(iv) In the action of G by conjugation on its set of subgroups, the G-orbit of H ≤ G is just the set of all conjugates of H in G, i.e.,{gHg−1 | g ∈ G}. The stabilizer of H in G is an important subgroup termed the normalizer of H in G,

NG(H )= {g ∈ G | gHg−1 = H}.

5.2 Orbits and stabilizers 83 Next we will prove two basic theorems on group actions. The first one counts the number of elements in an orbit.

(5.2.1) Let G be a group acting on a set X and let x ∈ X. Then the assignment

g StG(x) → g · x determines a bijection from the set of left cosets of StG(x) in G to

the orbit G· x. Hence |G · x| = |G : StG(x)|.

Proof. In the first place g StG(x) → g · x determines a well-defined function. For if s ∈ StG(x), then gs· x = g · (s · x) = g · x. Next g1· x = g2· x implies that

g−1₂ g1· x = x, so g₂−1g1 ∈ StG(x), i.e., g1StG(x) = g2StG(x). So the function is

injective, while it is obviously surjective.

Corollary (5.2.2) Let G be a finite group acting on a finite set X. If the action is

transitive, then|X| divides |G|, while if the action is regular, |X| = |G|.

Proof. If the action is transitive, X is the only G-orbit and so|X| = |G : StG(x)| for any x ∈ X, by (5.2.1); hence this divides |G|. If the action is regular, then in addition

StG(x)= 1 and thus |X| = |G|.

The corollary tells us that if G is a transitive permutation group of degree n, then

ndivides|G|, and |G| = n if G is regular.

The second main theorem on actions counts the number of orbits and has many applications. If a group G acts on a set X and g∈ G, the fixed point set of g is defined to be

Fix(g)= {x ∈ X | g · x = x}.

(5.2.3) (Burnside’s2Lemma). Let G be a finite group acting on a finite set X. Then

the number of G-orbits equals

1 |G|

 g∈G

|Fix(g)|,

i.e., the average number of fixed points of elements of G.

Proof. Consider how often an element x of X is counted in the sum_g∈G|Fix(g)|.

This happens once for each g in StG(x). Thus the element x contributes|StG(x)| = |G|/|G · x| to the sum, by (5.2.1). The same is true of each element of the orbit |G · x|, so that the total contribution of the orbit to the sum is

(|G|/|G · x|) · |G · x| = |G|.

It follows that_g∈G|Fix(g)| must equal |G| times the number of orbits, and the result

is proven.

We illustrate Burnside’s Lemma by a simple example.

Example (5.2.2) The group G= { id, (1 2)(3)(4), (1)(2)(3 4), (1 2)(3 4)} acts on the

set X = {1, 2, 3, 4} as a permutation group. There are two G-orbits here, namely {1, 2} and {3, 4}. Now it is easy to count the fixed points of the elements of G by looking for 1-cycles. Thus the four elements of the group have respective numbers of fixed points 4, 2, 2, 0. Therefore the number of G-orbits should be

1 |G|   g∈G |Fix(g)|= 1 4(4+ 2 + 2 + 0) = 2, which is the correct answer.

Example (5.2.3) Show that the average number of fixed points of an element of Sn is 1.

Here the symmetric group Snacts on the set{1, 2, . . . , n} in the natural way and the action is clearly transitive. By Burnside’s Lemma the average number of fixed points is simply the number of S_n-orbits, which is 1 by transitivity of the action.

Exercises (5.2)

1. If g is an element of a finite group G, the number of conjugates of g divides |G : g|.

2. If H is a subgroup of a finite group G, the number of conjugates of H divides |G : H |.

3. Let G = (1 2 . . . p), (1)(2 p)(3 p − 1) . . .  be a dihedral group of order 2p where p is an odd prime. Verify Burnside’s Lemma for the group G acting on the set {1, 2, . . . , p} as a permutation group.

4. Let G be a finite group acting as a finite set X. If the action is semiregular, prove that|G| divides |X|.

5. Prove that the class number of a finite group G is given by the formula 1 |G|   x∈G |CG(x)|  .

6. The class number of the direct product H × K equals the product of the class numbers of H and K.

7. Let G be a finite group acting transitively on a finite set X where|X| > 1. Prove that G contains at least|X| − 1 fixed-point-free elements, i.e., elements g such that Fix(g) is empty.

8. Let H be a proper subgroup of a finite group G. Prove that G=_x∈GxH x−1. [Hint: Consider the action of G on the set of left cosets of H by multiplication. The action is transitive, so Exercise (5.2.7) may be applied.]

5.3 Applications to the structure of groups 85