2 Characters of Finite Abelian Groups 401The functionχ 1 :G→Cdefined byχ 1 (a)=1foreverya∈Gis obviously a
character ofG,thetrivial character(also called theprincipalcharacter!). Moreover,
for any characterχofG, the functionχ−^1 :G→Cdefined byχ−^1 (a)=χ(a)−^1 is
also a character ofG.Furthermore,ifχ′andχ′′are characters ofG, then the function
χ′χ′′:G→Cdefined byχ′χ′′(a)=χ′(a)χ′′(a)is a character ofG.Since
χ 1 χ=χ, χ′χ′′=χ′′χ′,χ(χ′χ′′)=(χχ′)χ′′,it follows that the setGˆof all characters ofGis itself an abelian group, thedual group
ofG, with the trivial character as identity element.
Suppose now that the groupGis finite, of ordergsay. Thenχ(a)is ag-th root of
unity for everya∈G,sinceag=eand hence
χ(a)g=χ(ag)=χ(e)= 1.It follows that|χ(a)|=1andχ−^1 (a)=χ(a). Thus we will sometimes writeχ ̄
instead ofχ−^1.
Proposition 1The dual groupG of a finite abelian group G is a finite abelian groupˆ
of the same order. Moreover, if a∈G and a=e, thenχ(a)= 1 for someχ∈G.ˆ
Proof Letgdenote the order ofG. Suppose first thatGis a cyclic group, generated
by the elementc. Then any characterχofGis uniquely determined by the valueχ(c),
which is ag-th root of unity. Conversely ifωj=e^2 πij/g( 0 ≤j<g)is ag-th root
of unity, then the functionsχ(j):G→Cdefined byχ(j)(ck)=ωkj are distinct
characters ofGandχ(^1 )(ck)=1for1≤k<g. It follows that the proposition is true
whenGis cyclic. The general case can be reduced to this by using the fact (see§4of
Chapter III) that any finite abelian group is a direct product of cyclic groups. However,
it can also be treated directly in the following way.
We use induction ongand suppose thatGis not cyclic. LetHbe a maximal proper
subgroup ofGand lethbe the order ofH.Leta∈G\Hand letrbe the least positive
integer such thatb=ar∈H.SinceGis generated byHanda,andan∈Hif and
only ifrdividesn, eachx∈Gcan be uniquely expressed in the form
x=aky,wherey∈Hand 0≤k<r. Henceg=rh.
Ifχis any character ofG, its restriction toHis a characterψofH. Moreoverχ
is uniquely determined byψand the valueχ(a),since
χ(aky)=χ(a)kψ(y).Sinceχ(a)r=ψ(b)is a root of unity,ω=χ(a)is a root of unity such thatωr=ψ(b).
Conversely, it is easily verified that, for each characterψofHand for each of
therroots of unityωsuch thatωr =ψ(b), the functionχ :G→Cdefined by
χ(aky)=ωkψ(y)is a character ofG.SinceHhas exactlyhcharacters by the induc-
tion hypothesis, it follows thatGhas exactlyrh=gcharacters. It remains to show
that ifaky=e,thenχ(aky)=1forsomeχ.Butifωkψ(y)=1forallω,thenk=0;
hencey=eandχ(y)=ψ(y)=1forsomeψ, by the induction hypothesis.