Number Theory: An Introduction to Mathematics

5 Characters of Arbitrary Finite Groups 417

Sinceχv(e)=nvis the degree of thev-th irreducible representation, it follows from (7)
thatmv=nv. Thusevery irreducible representation is contained in the direct sum
decomposition of the regular representation, and moreover each occurs as often as its
degree.
It follows that
∑

μ

nμ^2 =g,

∑

μ

nμχμ(s)=0ifs=e. (9)

Thus the total number of functionsα(μ)ij is

∑

μnμ

(^2) =g. Therefore, since they are
linearly independent,every functionφ:G→Cis a linear combination of functions
αij(μ)occurring in irreducible matrix representations.
We show next thatevery class functionφ:G →Cis a linear combination of
irreducible charactersχμ. By what we have just provedφ=

∑

μφμ,where

φμ=

∑nμ

i,j= 1

λ(μ)ij αij(μ)

andλ(μ)ij ∈C.Butφ(st)=φ(ts)and

φμ(st)=

∑

i,j,k

λ(μ)ikαij(μ)(s)α(μ)jk(t), φμ(ts)=

∑

i,j,k

λ(μ)kjα(μ)ki (t)αij(μ)(s).

Since the functionsα(μ)ij are linearly independent, we must have

∑

k

λ(μ)ikα(μ)jk(t)=

∑

k

λ(μ)kjα(μ)ki (t).

If we denote byT(μ)the transpose of the matrix(λ(μ)ik), we can rewrite this in the form

A(μ)(t)T(μ)=T(μ)A(μ)(t).

Consequently, by Schur’s lemma,T(μ) = λμInμ and henceφμ = λμχμ. Thus
φ=

∑

μλμχμ.
Two e l e m e n t su,vof a groupGare said to beconjugateifv =s−^1 usfor some
s∈G. It is easily verified that conjugacy is an equivalence relation. ConsequentlyG
is the union of pairwise disjoint subsets, calledconjugacy classes, such that two ele-
ments belong to the same subset if and only if they are conjugate. The inverses of all
elements in a conjugacy class again form a conjugacy class, theinverse class.
In this terminology a functionφ :G →Cis a class function if and only if
φ(u)=φ(v)wheneveruandvbelong to the same conjugacy class. Thus the number
of linearly independent class functions is just the number of conjugacy classes inG.
Since the charactersχμform a basis for the class functions, it follows thatthe number
of inequivalent irreducible representations is equal to the number of conjugacy classes
in the group.
If a group of orderghasrconjugacy classes then, by (9),g=n^21 +···+n^2 r.
Since it is abelian if and only if every conjugacy class contains exactly one element,