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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
(a1 a2 ) = b a1 b a2 ,
b1 b2
a = b1 b2

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

The following equations are then satis¬ed by ν:
s(b1 )s(b2 ) = ν(b1 , b2 ) + s(b1 b2 )


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);
(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
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 ).
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 ),
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
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
element in H• (Q, A).

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

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 )
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 ).
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,


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


1 ’’ Z/2n ’’Qn+1 ’’ Z/2 ’’ 1
with 2-cocycles for which
ν(1, 1) = 0


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.

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
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
Let us ¬rst consider the set of all ˜square integrable™ functions T ’’ C, L2 (T). Such a
function f is required to have integrals such as ’π |f (eti )|2 dt. There is an inner product on
L2 (T),
f (eti )g(eti ) dt
(f |g) =
2π ’π
which is Hermitian. Notice that
|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 .

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

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
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.


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