apart from the identity element every element of this group is of order 2. But the elements of

Q8 ’ {±1} are all of order 4, hence there can be no such subgroup H.

3. Classifying extensions [optional extra material ]

We will now discuss the classi¬cation of extensions of N by Q for a given action • : Q ’’

Aut(N ). To ease the technical details we will assume that N = A is an abelian group. We will

continue to use the notation b a = •(b)(a) for a ∈ Q and a ∈ A. Using properties of the action

• we obtain the identities

b

(a1 a2 ) = b a1 b a2 ,

b1 b2

a = b1 b2

a.

So suppose that we have an extension

j π

1 ’’ A ’ G ’ Q ’’ 1

’ ’

with action • : Q ’’ Aut(A). We will view j as the inclusion of a normal subgroup and also

interpret Q as the quotient group G/A with π the projection given by π(g) = gA.

Suppose that we have a function s : Q ’’ G with the following properties:

(i) π(s(x)) = x (x ∈ Q);

(ii) s(eQ ) = eG .

The function s is called a section and its image im s is called a transversal of A in G. Since G

is the disjoint union of the distinct cosets Ag = gA and these have empty intersection, the ¬rst

of these conditions amounts to the fact that any g ∈ G has a unique expression as

g = as(π(g)) (a ∈ A).

For a ∈ A and b ∈ Q we have

b

a = s(b)as(b)’1 .

Notice that this is independant of the actual function s used, since A is abelian and hence for

a second section s , we have s (b)s(b)’1 ∈ A, giving

s (b)as (b)’1 = (s (b)s(b)’1 )s(b)as(b)’1 (s (b)s(b)’1 ) = s(b)as(b)’1 .

For b1 , b2 ∈ Q we also have

s(b1 )s(b2 ) ∈ As(b1 b2 )

since s(b1 b2 ) ∈ Ab1 b2 = Ab1 Ab2 . Hence we can de¬ne a function ν : Q — Q ’’ A by the

equation

ν(b1 , b2 ) = s(b1 )s(b2 )s(b1 b2 )’1 .

66 6. AUTOMORPHISMS AND EXTENSIONS

The following equations are then satis¬ed by ν:

s(b1 )s(b2 ) = ν(b1 , b2 ) + s(b1 b2 )

and

ν(eQ , b) = eA = ν(b, eQ ) (b ∈ Q).

A function f : Q — Q ’’ A is called a factor set or 2-cocycle with respect to the action •

if the following hold.

(CoCy-1) f (eQ , x) =0 = f (x, eQ ) (b ∈ Q);

x

(CoCy-2) f (y, z) + f (x, yz) = f (xy, z) + f (x, y) (x, y, z ∈ Q).

Then the function ν discussed above is a 2-cocycle. Given two 2-cocycles f1 , f2 the function

f1 · f2 de¬ned by

(f1 · f2 )(x, y) = f1 (x, y)f2 (x, y)

is easily be veri¬ed to be another 2-cocycle (this uses the fact that A is abelian). In fact, the

2

set of all 2-cycles Z• (Q, A) (with respect to the given action •) becomes an abelian group,

with identity element the constant function taking the value eA , and the inverse of f being the

function f (’1) for which

f (’1) (b1 , b2 ) = f (b1 , b2 )’1 .

A 2-cocycle f is a 2-coboundary with respect to the action • if there is a function F : Q ’’ A

for which

f (b1 , b2 ) = b1 F (b2 )F (b1 b2 )’1 F (b1 ) = δF (b1 , b2 ).

(CoCy)

2 2

The set of all 2-cocyles B• (Q, A) also a forms an abelian group which is a subgroup of Z• (Q, A).

We de¬ne the 2nd cohomology group with respect to the action • to be the quotient group

2 2 2

H• (Q, A) = Z• (Q, A)/B• (Q, A).

To see why 2-coboundaries are of interest, consider what happens if we change the section s

above to another s . Then the function F : Q ’’ A given by

F (b) = s (b)s(b)’1 ∈ A

and the 2-cocycle ν associated with s satisfy

ν (b1 , b2 ) = δF (b1 , b2 )ν(b1 , b2 ),

2

hence they represent the same element of H• (Q, A).

Theorem 6.16. Let • : Q ’’ Aut(A) be an action with A abelian.

j q

Let 1 ’’ A ’ G ’’ Q ’ 1 be an extension with associated action • of Q on A. Then the

’ ’

2 2

coset ν B• (Q, A) ∈ H• (Q, A) is independent of the section s used to de¬ne it. This element of

2

H• (Q, A) only depends on the extension up to equivalence of extensions.

2 2

Conversely, given an element ν B• (Q, A) ∈ H• (Q, A), any representative ν de¬nes an

j q

extension 1 ’’ A ’ G ’’ Q ’ 1, which is unique up to equivalence of extensions.

’ ’

These constructions set up a bijection

j π 2

equivalence classes of extensions 1 ’’ A ’ G ’ Q ’’ 1

’ ’ elements of H• (Q, A).

Corollary 6.17. In this correspondence, the split extensions correspond to the identity

2

element in H• (Q, A).

Proof. For a split extension we can take ν(b1 , b2 ) = eA .

3. CLASSIFYING EXTENSIONS [OPTIONAL EXTRA MATERIAL] 67

Theorem 6.18. Let A and G be ¬nite groups with A abelian. If |A| = m and |Q| = n with

m, n coprime, i.e., (m, n) = 1, then any extension of A by Q is split.

Proof. We give an outline of the proof.

First note that in A the equation xn = a always has a solution, since m and n are coprime,

hence for some r, s ∈ Z, rm + sn = 1 in A. As |a| | |A| we have a = arm asn = (as )n . This

solution is unique since if xn = y n = a then (yx’1 )n = eA and as |yx’1 | | m, this implies y = x.

Let ν be a two cocycle. For each b ∈ Q there is a unique solution in A of the equation

xn = ν(b, b ),

b ∈Q

hence there is a unique function F : Q ’’ A for which

F (b)n = ν(b, b ).

b ∈Q

Now we have

δF (b1 , b2 ) = ν(b1 , b2 )

2

and hence ν ∈ B• (Q, A), which implies that the extension is trivial.

A more general version of this is contained in the following result for which a proof can be

found in [A&B], chapter 3, §9.

A subgroup H G is called a Hall subgroup if |H| and |G : H| are coprime. If N G is a

normal subgroup, then a subgroup Q G for which G = N Q is called a complement of N

in G (notice that G is then a split extension of N by Q).

Theorem 6.19 (Schur“Zassenhaus Theorem). Let G be ¬nite group with a normal Hall

subgroup N G. Then N has a complement Q G in G. Moreover, any other such complement

Q is conjugate to Q in G.

We end with some information about extensions of the form

1 ’’ Z/2n ’’ G ’’ Z/2 ’’ 1

for n 1. Let µ : Z/2 ’’ Aut(Z/2n ) denote the trivial action and „ : Z/2 ’’ Aut(Z/2n ) the

action given by

„ (1)(t) = ’t.

Lemma 6.20. We have

Hµ (Z/2, Z/2n ) ∼ Z/2 ∼ H„ (Z/2, Z/2n ).

2

=2

=

For the case trivial action, it is easy to see that two inequivalent extensions are

1 ’’ Z/2n ’’Z/2n — Z/2 ’’ Z/2 ’’ 1,

and

1 ’’ Z/2n ’’Z/2n+1 ’’ Z/2 ’’ 1

and these can be formed with the 2-cocycles for which

ν(1, 1) = 0,

ν(1, 1) = 2.

For the action „ , we have extensions

1 ’’ Z/2n ’’D2n+1 — Z/2 ’’ Z/2 ’’ 1

68 6. AUTOMORPHISMS AND EXTENSIONS

and

1 ’’ Z/2n ’’Qn+1 ’’ Z/2 ’’ 1

with 2-cocycles for which

ν(1, 1) = 0

and

ν(1, 1) = 2n’1 .

The ¬rst type of group is a dihedral group of order 2n+1 , while the second is a quaternion group

of order 2n+1 . If n = 2, these are the familiar dihedral and quaternion groups.

CHAPTER 7

Some further applications

In this chapter we will see some other applications of representation theory in mathematics

and in other subject areas.

1. Fourier series and the circle group

A ˜reasonable™ function f : R ’’ C which is periodic of period 2π can be expended in a

Fourier series

∞

c eti ,

F (t) = n

’∞

where the Fourier coe¬cients cn ∈ C are given by

π

1

f (t)e’ti dt.

cn =

2π ’π

˜

Such a function can be viewed as de¬ning a function f : T ’’ C on the unit circle

T = {z ∈ C : |z| = 1}

which is an abelian group under multiplication (in fact, the multiplication and inverse maps are

continuous and even di¬erentiable functions). This function is given by

˜

f (eti ) = f (t).

A natural question to ask is whether there is a some sort of representation theory for such a

group.

Let us ¬rst consider the set of all ˜square integrable™ functions T ’’ C, L2 (T). Such a

π

function f is required to have integrals such as ’π |f (eti )|2 dt. There is an inner product on

L2 (T),

π

1

f (eti )g(eti ) dt

(f |g) =

2π ’π

which is Hermitian. Notice that

π

1

|f (eti )|2 dt 0,

(f |f ) =

2π ’π

with equality if and only if f = 0. Since T is abelian, L2 (T) can be thought of as the set of

class functions on T. However, unlike the case of a ¬nite group, L2 (T) is an in¬nite dimensional

vector space over C (in fact it is a Hilbert space). The functions χn (t) = eti (n ∈ Z) form an

orthonormal set since

(χm |χn ) = δm,n .

Moreover, since any function in L2 (T) has a Fourier expansion, the χn actually span in the sense

that every f ∈ L2 (T) can be expressed in the form

∞

f= cn χn .

n=’∞

Thus the χn form an orthonormal basis for L2 (T) (in the sense of Hilbert space theory).

69

70 7. SOME FURTHER APPLICATIONS

What about representations of T? For each n ∈ Z, there is a (continuous) 1-dimensional

representation ρn : T ’’ C— given by

ρn (z) = z n .

The character of this is the function χn . In fact, these ρn are the only irreducible (continuous)

representations of T. It can be shown that any ˜reasonable™ representation can be decomposed

into something like a direct sum of copies of the representations ρn . Moreover, the rˆle of the

o

regular representation is played by the vector space L2 (T) on which T acts by

w · f (z) = f (w’1 z) (z, w ∈ T).

The multiplicity of ρn in L2 (T) is 1.

Thus Fourier expansions are intimately related to representation theory. In fact, any com-

pact subgroup G of a group GLC (Cn ) is a compact Lie group and there is similarly a good

theory of continuous representations of G. This is the subject of Harmonic analysis on compact

Lie groups. It can even be extended to noncompact groups such as GLC (Cn ) and is a major

area of research.