Number Theory: An Introduction to Mathematics

428 X A Character Study

It has been shown by Kazarin (1990) that the normal subgroup generated by the
elements of the conjugacy classCin Proposition 18 issolvable. Although no proof of
Burnside’s Proposition 18 is known which does not use character theory, Goldschmidt
(1970) and Matsuyama (1973) have given a rather intricate proof of the important
Corollary 19 which is purely group theoretic.
The restriction totwodistinct primes in the statement of Corollary 19 is essential,
since the alternating groupA 5 of order 60= 22 · 3 ·5 is simple. It follows at once from
Corollary 19, by induction on the order, that any finite group whose order is divisible
by at most two distinct primes issolvable. P. Hall (1928/1937) has used Corollary 19 to
show that a finite groupGof ordergis solvable if and only ifGhas a subgroupHof
orderhfor every factorizationg=pah,wherea>0andpis a prime not dividingh.
The second application of group characters, due to Frobenius, has the following
statement:

Proposition 20If the finite group G has a nontrivial proper subgroup H such that

x−^1 Hx∩H={e} for every x∈G\H,

then G contains a normal subgroup N such that G is the semidirect product of H and
N, i.e.

G=NH, H∩N={e}.

Proof Obviouslyx−^1 Hx= y−^1 Hyif y ∈ Hxand the hypotheses imply that
x−^1 Hx∩y−^1 Hy={e}ify ∈/Hx.Ifg,hare the orders ofG,Hrespectively, it
follows that the number of distinct conjugate subgroupsx−^1 Hx(includingHitself) is
n=g/h. Furthermore the number of elements ofGwhich belong to some conjugate
subgroup isn(h− 1 )+ 1 =g−(n− 1 ). Thus the setSof elements ofGwhich do
not belong to any conjugate subgroup has cardinalityn−1.
Letψμbe the character of an irreducible representation ofHandψ ̃μthe character
of the induced representation ofG. By (12) and the hypotheses,

ψ ̃μ(e)=nψμ(e), ψ ̃μ(s)=0ifs∈S, ψ ̃μ(s)=ψμ(s) ifs∈H\e.

For any fixedμ, form the class function

χ=ψ ̃μ−ψμ(e){ψ ̃ 1 −χ 1 },

whereψ 1 andχ 1 are the characters of the trivial representations ofHandGrespec-
tively. Thenχis ageneralized characterofG,i.e.χ =

∑

vmvχvis a linear com-
bination of irreducible charactersχvwith integral, but not necessarily non-negative,
coefficientsmv. Moreover

χ(e)=ψμ(e), χ(s)=ψμ(e) ifs∈S,χ(s)=ψμ(s) ifs∈H\e.

Hence
∑

s∈H\e

χ(s)χ(s−^1 )=

∑

s∈H\e

ψμ(s)ψμ(s−^1 )=h−ψμ(e)^2.