Number Theory: An Introduction to Mathematics

7 Applications 429

SinceShas cardinalityn−1, it follows that
∑

s∈G

χ(s)χ(s−^1 )=n{h−ψμ(e)^2 }+ψμ(e)^2 +(n− 1 )ψμ(e)^2 =g.

But the formula (8) holds also for generalized characters. Sinceχ(e)>0, we conclude
thatχis in fact an irreducible character ofG. Thus we have an irreducible representa-
tion of degreeχ(e) in which the matrices representing elements ofShave traceχ(e).
The elements ofSmust therefore be represented by the unit matrix, i.e. they belong to
the kernelKμof the representation.
On the other hand, for anyt∈H\ewe have
∑

μ

ψμ(e)ψμ(t)= 0

and henceψμ(t)=ψμ(e)for someμ. Thus the intersection of the kernelsKμfor
varyingμcontains just the elements ofSande.SinceKμis a normal subgroup ofG,
it follows thatN=S∪{e}is also a normal subgroup. Furthermore, sinceH∩N={e},
HNhas cardinalityhn=gand henceHN=G.

A finite groupGwhich satisfies the hypotheses of Proposition 20 is said to be a
Frobenius group. The subgroupHis said to be aFrobenius complementand the normal
subgroupNaFrobenius kernel. It is readily shown that a finite permutation group is a
Frobenius group if and only if it is transitive and no element except the identity fixes
more than one symbol. Another characterization follows from Proposition 20: a finite
groupGis a Frobenius group if and only if it has a nontrivial proper normal subgroup
Nsuch that, ifx∈Nandx=e,thenxy=yxfor ally∈G\N.
Frobenius groups are of some general significance and much is known about their
structure. It is easily seen thathdividesn−1, so that the subgroupsHandNhave rel-
atively prime orders. It has been shown by Thompson (1959) that the normal subgroup
Nis a direct product of groups of prime power order. The structure ofHis known
even more precisely through the work of Burnside (1901) and others.
Applications of group characters of quite a different kind arise in the study of mole-
cular vibrations. We describe one such application within classical mechanics, due to
Wigner (1930). However, there are further applications within quantum mechanics,
e.g. to the determination of the possible spectral lines in the Raman scattering of light
by a substance whose molecules have a particular symmetry group.
A basic problem of classical mechanics deals with thesmall oscillationsof a sys-
tem of particles about an equilibrium configuration. The equations of motion have the
form
Bx ̈+Cx= 0 , (13)

wherex∈Rnis a vector of generalized coordinates andB,Care positive definite real
symmetric matrices. In fact the kinetic energy is( 1 / 2 )x ̇tBx ̇and, as a first approxima-
tion forxnear 0, the potential energy is( 1 / 2 )xtCx.
SinceBandCare positive definite, there exists (see Chapter V,§4) a non-singular
matrixTsuch that

TtBT=I, TtCT=D,