1.2 The Concept of a Group Action

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Let X be a set, and let G be a group. Say that G acts on X if there is a

homomorphism _ : G ! SX. (The homomorphism _ : G ! SX is sometimes

referred to as a group action .) It is customary to write gx or g · x in place

of _(g)(x), when g 2 G, x 2 X. In the last section we already met the

prototypical example of a group action. Indeed, if G is a group and H _ G

then there is a homomorphsm G ! SG/H, i.e., G acts on the quotient set

G/H by left multiplication. If K = ker_ we say that K is the kernel of the

action. If this kernel is trivial, we say that the group acts faithfully on X,

or that the group action is faithful .

Let G act on the set X, and let x 2 X. The stabilizer , StabG(x), of x

in G, is the subgroup

StabG(x) = {g 2 G| g · x = x}.

Note that StabG(x) is a subgroup of G and that if g 2 G, x 2 X, then

StabG(gx) = gStabG(x)g−1. If x 2 X, the G-orbit in X of x is the set

OG(x) = {g · x| g 2 G} _ X.

If g 2 G set

Fix(g) = {x 2 X| g · x = x} _ X,

the fixed point set of g in X. More generally, if H _ G, there is the set of

H-fixed points :

Fix(H) = {x 2 X| h · x = x for all h 2 H}.

The following is fundamental.

Theorem 1.2.1 (Orbit-Stabilizer Reciprocity Theorem) Let G be a

finite group acting on the set X, and fix x 2 X. Then

|OG(x)| = [G : StabG(x)].

The above theorem is often applied in the following context. That is, let

G be a finite group acting on itself by conjugation (g · x = gxg−1, g, x 2 G).

In this case the orbits are called conjugacy classes and denoted

CG(x) = {gxg−1| g 2 G}, x 2 G.

6 CHAPTER 1. GROUP THEORY

In this context, the stabilizer of the element x 2 G, is called the centralizer

of x in G, and denoted

CG(x) = {g 2 G| gxg−1 = x}.

As an immediate corollary to Theorem 1.2.1 we get

Corollary 1.2.1.1 Let G be a finite group and let x 2 G. Then |CG(x)| =

[G : CG(x)].

Note that if G is a group (not necessarily finite) acting on itself by

conjugation, then the kernel of this action is the center of the group G:

Z(G) = {z 2 G| zxz−1 = x for all x 2 G}.

Let p be a prime and assume that P is a group (not necessarily finite)

all of whose elements have finite p-power order. Then P is called a p-group.

Note that if the p-group P is finite then |P| is also a power of p by Cauchy’s

Theorem.

Lemma 1.2.2 (“p on p0 ” Lemma) Let p be a prime and let P be a finite

p-group. Assume that P acts on the finite set X of order p0, where p 6 | p0.

Then there exists x 2 X, with gx = x for all g 2 P.

The following is immediate.

Corollary 1.2.2.1 Let p be a prime, and let P be a finite p-group. Then

Z(P) 6= {e}.

The following is not only frequently useful, but very interesting in its

own right.

Theorem 1.2.3 (Burnside’s Theorem) Let G be a finite group acting

on the finite set X. Then

1

|G|

X

g2G

|Fix(g)| = # of G-orbits in X.

Burnside’s Theorem often begets amusing number theoretic results. Here

is one such (for another, see Exercise 4, below):

1.2. THE CONCEPT OF A GROUP ACTION 7

Proposition 1.2.4 Let x, n be integers with x _ 0, n > 0. Then

nX−1

a=0

x(a,n) _ 0 (mod n),

where (a, n) is the greatest common divisor of a and n.

Let G act on the set X; if OG(x) = X, for some x 2 X then G is

said to act transitively on X, or that the action is transitive . Note that

if G acts transitively on X, then OG(x) = X for all x 2 X. In light of

Burnside’s Theorem, it follows that if G acts transitively on the set X, then

the elements of G fix, on the average, one element of X.

There is the important notion of equivalent permutation actions. Let

G be a group acting on sets X1, X2. A mapping _ : X1 ! X2 is called

G-equivariant if for each g 2 G the diagram below commutes:

X1

_ - X2

X1

g

? _ - X2

g

?

If the G-equivariant mapping above is a bijection, then we say that the

actions of G on X1 and X2 are permutation isomorphic, .

An important problem of group theory, especially finite group theory, is

to classify, up to equivalence, the transitive permutation representations of

a given group G. That this is really an “internal” problem, can be seen from

the following important result.

Theorem 1.2.5 Let G act transitively on the set X, fix x 2 X, and set

H = StabG(x). Then the actions of G on X and on G/H are equivalent.

Thus, classifying the transitive permutation actions of the group G is

tantamount to having a good knowledge of the subgroup structure of G.

(See Exercises 5, 6, 8, below.)

8 CHAPTER 1. GROUP THEORY

Exercises 1.2

1. Let G be a group and let x, y 2 G. Prove that x and y are conjugate if

and only if there exist elements u, v 2 G such that x = uv and y = vu.

2. Let G be a finite group acting transitively on the set X. If |X| 6= 1

show that there exist elements of G which fix no elements of X.

3. Use Exercise 2 to prove the following. Let G be a finite group and let

H < G be a proper subgroup. Then G 6= [g2GgHg−1.

4. Let n be a positive integer, and let d(n) =# of divisors of n. Show

that

nX−1

a = 0

(a, n) = 1

(a − 1, n) = _(n)d(n),

where _ is the Euler _-function. (Hint: Let Zn = hxi be the cyclic

group of order n, and let G = Aut(Zn).2 What is |G|? [See Section

4, below.] How many orbits does G produce in Zn? If g 2 G has the

effect x 7! xa, what is |Fix(g)|?)

5. Assume that G acts transitively on the sets X1,X2. Let x1 2 X1, x2 2

X2, and let Gx1 ,Gx2 be the respective stabilizers in G. Prove that

these actions are equivalent if and only if the subgroups Gx1 and Gx2

are conjugate in G. (Hint: Assume that for some _ 2 G we have

Gx1 = _Gx2_−1. Show that the mapping _ : X1 ! X2 given by

_(gx1) = g_ (x2), g 2 G, is a well-defined bijection that realizes an

equivalence of the actions of G. Conversely, assume that _ : X1 ! X2

realizes an equivalence of actions. If y1 2 X1 and if y2 = _(x1) 2 X2,

prove that Gy1 = Gy2 . By transitivity, the result follows.)

6. Using Exercise 5, classify the transitive permutation representations

of the symmetric group S3.

7. Let G be a group and let H be a subgroup of G. Assume that H =

NG(H). Show that the following actions of G are equivalent:

(a) The action of G on the left cosets of H in G by left multiplication;

2For any group G, Aut(G) is the group of all automorphisms of G, i.e. isomorphisms

G ! G. We discuss this concept more fully in Section 1.5.

1.2. THE CONCEPT OF A GROUP ACTION 9

(b) The action of G on the conjugates of H in G by conjugation.

8. Let G = ha, bi _= Z2 × Z2. Let X = {±1}, and let G act on X in the

following two ways:

(a) aibj · x = (−1)i · x.

(b) aibj · x = (−1)j · x.

Prove that these two actions are not equivalent.

9. Let G be a group acting on the set X, and let N / G. Show that G

acts on Fix(N).

10. Let G be a group acting on a set X. We say that G acts doubly

transitively on X if given x1 6= x2 2 X, y1 6= y2 2 X there exists

g 2 G such that gx1 = y1, gx2 = y2.

(i) Show that the above condition is equivalent to G acting transitively

on X × X − _(X × X), where G acts in the obvious way

on X × X and where _(X × X) is the diagonal in X × X.

(ii) Assume that G is a finite group acting doubly transitively on the

set X. Prove that

1

|G|

X

g2G

|Fix(g)|2 = 2.

11. Let X be a set and let G1,G2 _ SX. Assume that g1g2 = g2g1 for all

g1 2 G1, g2 2 G2. Show that G1 acts on the G2-orbits in X and that

G2 acts on the G1-orbits in X. If X is a finite set, show that in the

above actions the number of G1-orbits is the same as the number of

G2-orbits.

12. Let G act transitively on the set X via the homomorphism _ : G ! SX,

and define Aut(G,X) = CSX(G) = {s 2 SX| s_(g)(x) = _(g)s(x) for all g 2

G}. Fix x 2 X, and let Gx = StabG(x). We define a new action of

N = NG(Gx) on X by the rule n _ (g · x) = (gn−1) · x.

(i) Show that the above is a well defined action of N on X.

(ii) Show that, under the map n 7! n_, n 2 N, one has N !

Aut(G,X).

(iii) Show that Aut(G,X) _= N/Gx. (Hint: If c 2 Aut(G,X), then

by transitivity, there exists g 2 G such that cx = g−1x. Argue

that, in fact, g 2 N, i.e., the homomorphism of part (ii) is onto.)

10 CHAPTER 1. GROUP THEORY

13. Let G act doubly transitively on the set X and let N be a normal

subgroup of G not contained in the kernel of the action. Prove that

N acts transitively on X. (The double transitivity hypothesis can be

weakened somewhat; see Exercise 15 of Section 1.6.)

14. Let A be a finite abelian group and define the character group A_ of

A by setting A_ = Hom(A,C×), the set of homomorphisms A ! C×,

with pointwise multiplication. If H is a group of automorphisms of A,

then H acts on A_ by h(_)(a) = _(h(a−1)), _ 2 A_, a 2 A, h 2 H.

(a) Show that for each h 2 H, the number of fixed points of h on A

is the same as the number of fixed points of h on A_.

(b) Show that the number of H-orbits in A equals the number of

H-orbits in A_.

(c) Show by example that the actions of H on A and on A_ need not

be equivalent.

(Hint: Let A = {a1, a2, . . . , an}, A_ = {_1, _2, . . . , _n} and form

the matrix X = [xij ] where xij = _i(aj ). If h 2 H, set P(h) =

[pij ], Q(h) = [qij ], where

pij =

n 1 if h(_i) = _j

0 if h(_i) 6= _j

, qij =

n 1 if h(ai) = aj

0 if h(ai) 6= aj

.

Argue that P(h)X = XQ(h); by Exercise 9 of page 4 one has that

X ·X_ = |A| · I, where X_ is the onjugate transpose of the matrix X.

In particular, X is nonsingular and so trace P(h) = trace Q(h).)

15. Let G be a group acting transitively on the set X, and let _ : G ! G

be an automorphism.

(a) Prove that there exists a bijection _ : X ! X such that _(g ·x) =

_(g)·_(x), g 2 G, x 2 X if and only if _ permutes the stabilizers

of points x 2 X.

(b) If _ : X ! X exists as above, show that the number of such

bijections is [NG(H) : H], where H = StabG(x), for some x 2

X. (If the above number is not finite, interpret it as a cardinality.)

16. Let G be a finite group of order n acting on the set X. Assume the

following about this action:

1.2. THE CONCEPT OF A GROUP ACTION 11

(a) For each x 2 X, StabG(x) 6= {e}.

(b) Each e 6= g 2 G fixes exactly two elements of X.

Prove that X is finite; if G acts in k orbits on X, prove that one of

the following must happen:

(a) |X| = 2 and that G acts trivially on X (so k = 2).

(b) k = 3.

In case (b) above, write k = k1 + k2 + k3, where k1 _ k2 _ k3 are the

sizes of the G-orbits on X. Prove that k1 = n/2 and that k2 < n/2 implies

that n = 12, 24 or 60. (This is exactly the kind of analysis needed

to analyize the proper orthogonal groups in Euclidean 3-space; see e.g.,

L.C. Grove and C.T. Benson, Finite Reflection Groups”, Second ed.,

Springer-Verlag, New York, 1985, pp. 17-18.)

12 CHAPTER 1. GROUP THEORY

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