Number Theory: An Introduction to Mathematics

7 Applications 425

self-conjugate characterψ 4 ofA 4 yields two irreducible charactersχ 4 ,χ 5 ofS 4 of
degree 3. The rows of the character table corresponding toχ 4 ,χ 5 must have the form

3 xzw y

3 −xzw −y

and from the orthogonality relations (11) we obtainz=−1,w=0,xy=−1. From
the orthogonality relations (10) we further obtainx+y=0. Hencex^2 =1andthe
complete character table is

S 4

|C| 16 38 6

CC 1 C 2 C 3 C 4 C 5

χ 1 1 1111

χ 2 1 − 111 − 1

χ 3 202 − 10

χ 4 31 − 10 − 1

χ 5 3 − 1 − 101

The physical significance of these groups derives from the fact thatA 4 (resp.S 4 )
is isomorphic to the group of all rotations (resp. orthogonal transformations) ofR^3
which map a regular tetrahedron onto itself. SimilarlyA 3 (resp.S 3 ) is isomorphic to
the group of all plane rotations (resp. plane rotations and reflections) which map an
equilateral triangle onto itself.
An important property of induced representations was proved by R. Brauer (1953):
each character of a finite group is a linear combination with integer coefficients (not
necessarily non-negative) of characters induced from characters of elementary sub-
groups. Here a group is said to beelementaryif it is the direct product of a group
whose order is a power of a prime and a cyclic group whose order is not divisible by
that prime.
It may be deduced without difficulty from Brauer’s theorem that, ifGis a finite
group andmthe least common multiple of the orders of its elements, then (as had long
been conjectured) any irreducible representation ofGis equivalent to a representation
in the fieldQ(e^2 πi/m). Green (1955) has shown that Brauer’s theorem is actually best
possible: if each character of a finite groupGis a linear combination with integer coef-
ficients of characters induced from characters of subgroups belonging to some family
F, then each elementary subgroup ofGis contained in a conjugate of some subgroup
inF.

7 Applications................................................

Character theory has turned out to be an invaluable tool in the study of abstract groups.
We illustrate this by two results of Burnside (1904) and Frobenius (1901). It is remark-
able, first that these applications were found so soon after the development of character
theory and secondly that, one century later,there are still no proofs known which do
not use character theory.