Advanced High-School Mathematics

240 CHAPTER 4 Abstract Algebra

12. Let R be the additive group of real numbers and assume that
    f :R→Ris a function which satisfies the peculiar property that
    f(x^2 −y^2 ) =xf(x)−yf(y) for allx, y∈R.
    (a) Prove thatf:R→Ris a homomorphism, and that
    (b) there exists a real numberc∈Rsuch thatf(x) =cx, x∈R.
    The result of Exercise 12 is strictly stronger that that of Exer-
    cise 11. Indeed the condition of Exercise 12 shows that the homo-
    morphism is continuous and of a special form. We’ll return to
    this briefly in Chapter 5; see Exercise 6 on page 252.


13. Let G be a group and let C∗ denote the multiplicative group of
    complex numbers. By a (linear)characterwe mean a homomor-
    phismχ :G → C∗, i.e., χ(g 1 g 2 ) =χ(g 1 )χ(g 2 ) for all g 1 , g 2 ∈G.

(a) Prove that ifχ:G→C∗is a character, then χ(g−^1 ) =χ(g)

(complex conjugate) for allg∈G.

Now assume thatGis finite and that χ:G →C∗ is a character

such that for at least oneg∈G,χ(g) 6 = 1. Prove that

(b)

∑

g∈G

χ(g) = 0.

(c) Let∑ χ 1 , χ 2 : G → C∗ be distinct characters and prove that

g∈G

χ 1 (g)χ 2 (g) = 0.

(d) Fix the positive integer nand show that for any integerk =

0 , 1 , 2 ,...,n−1 the mappingχk:Zn→C∗given byχk(a) =

cos(2πka/n) +isin(2πka/n) is a character. Show that any

character ofZnmust be of the formχk, 0 ≤k < n, as above.

4.2.9 Return to the motivation

We return to the two graphs having six vertices each on page 206, and
make a simple observation about their automorphism groups. Namely,