Number Theory: An Introduction to Mathematics

418 X A Character Study

i.e. if and only ifr=g, it follows thata finite group is abelian if and only if every
irreducible representation has degree1.
LetC 1 ,...,Crbe the conjugacy classes of the groupGand lethkbe the number
of elements inCk(k= 1 ,...,r). Changing notation, we will now denote byχikthe
common value of the character of all elements in thek-th conjugacy class in thei-th
irreducible representation. Then, sinceχ(s−^1 )=χ(s), the orthogonality relations (6)
can be rewritten in the form

g−^1

∑r

j= 1

hjχijχkj=

{

1ifi=k,

0ifi=k.

(10)

Thus ther×rmatricesA=(χik),B=(g−^1 hiχki)satisfyAB=I. Therefore also
BA=I,i.e.

∑r

j= 1

χjiχjk=

{

g/hk ifi=k,

0ifi=k.

(11)

It may be noted thathkdividesgsince, for anysk∈Ck,g/hkis the order of the
subgroup formed by all elements ofGwhich commute withsk. We are going to show
finally thatthe degree of any irreducible representation divides the order of the group.
Any representationρ:s→A(s)of a finite groupGmay be extended by linearity
to the set of all linear combinations of elements ofG:

ρ

(∑

s∈G

αss

)

=

∑

s∈G

αsA(s).

In particular, letCkdenote the sum of all elements in thek-th conjugacy classCkofG.
For anyt,u∈G,

u−^1 skut=t(t−^1 u−^1 skut)

and hence

ρ(Ck)A(t)=

∑

s∈Ck

A(st)=

∑

s∈Ck

A(ts)=A(t)ρ(Ck).

Ifρ = ρi is an irreducible representation, it follows from Schur’s lemma that
ρi(Ck)=λikIni. Moreover, since

trρi(Ck)=hkχik,

wherehkagain denotes the number of elements inCk,wemusthaveλik=hkχik/ni.
Now let

C=

∑r

k= 1

(g/hk)CkCk′,

where Ck′ is the conjugacy class inverse to Ck. (Otherwise expressed,
C=

∑

s,t∈Gsts

− (^1) t− (^1) ).Thenρi(C)=γiIn
i,where
γi=
∑r
k= 1
(g/hk)λikλik=(g/n^2 i)
∑r
k= 1
hkχikχik=(g/ni)^2 ,