9 Further Remarks 4439 FurtherRemarks
The history of Legendre’s conjectures on primes in arithmetic progressions is
described in Vol. I of Dickson [13]. Dirichlet’s original proof is contained in [33],
pp. 313–342. Although no simple general proof of Dirichlet’s theorem is known,
simple proofs have been given for the existence of infinitely many primes congruent
to1modm; see Sedrakian and Steinig [41].
If all arithmetic progressionsa,a+m,...with(a,m) = 1 contain a prime,
then they all contain infinitely many, since for anyk>1 the arithmetic progression
a+mk,a+ 2 mk,...contains a prime.
It may be shown that any finite abelian groupGisisomorphicto its dual groupGˆ
(although not in a canonical way) by expressingGas a direct product of cyclic groups;
see, for example, W. & F. Ellison [15].
In the final step of the proof of Proposition 7 we have followed Bateman [3]. Other
proofs thatL( 1 ,χ)=0foreveryχ=χ 1 , which do not use Proposition 6, are given
in Hasse [21]. The functional equation for DirichletL-functions was first proved by
Hurwitz (1882). For proofs of some of the results stated at the end of§3, see Bach and
Sorenson [1], Davenport [12], W. & F. Ellison [15] and Prachar [40]. Funakura [18]
characterizes DirichletL-functions by means of their analytic properties.
The history of the theory of group representations and group characters is de-
scribed in Curtis [10]. More complete expositions of the subject than ours are given by
Serre [42], Feit [16], Huppert [27], and Curtis and Reiner [11]. The proof given here
that the degree of an irreducible representation divides the order of the group is not
Frobenius’ original proof. It first appeared in a footnote of a paper by Schur (1904) on
projective representations, where it is attributed to Frobenius. Zassenhaus [50] gives
an interpretation in terms ofCasimir operators.
A character-free proof of Corollary 19 is given in Gagen [19]. P. Hall’s theorem is
proved in Feit [16], for example. Frobenius groups are studied further in Feit [16] and
Huppert [27].
For physical and chemical applications of group representations, see Cornwell [9],
Janssen [29], Meijer [36], Birman [4] and Wilsonet al.[48].
Dym and McKean [14] give an outward-looking introduction to the classical theory
of Fourier series and integrals. The formal definition of a topological group is due to
Schreier (1926). The Haar integral is discussed by Nachbin [37]. General introductions
to abstract harmonic analysis are given by Weil [46], Loomis [34] and Folland [17].
More detailed information on topological groups and their representations is contained
in Pontryagin [39], Hewitt and Ross [23] and Gurarii [20]. A simple proof that the ad-
ditive groupQpof allp-adic numbers is isomorphic to its dual group is given by
Washington [45]. In the adelic approach to algebraic number theory this isomorphism
lies behind the functional equation of the Riemann zeta function; see, for example,
Lang [31].
For Hilbert’s fifth problem, see Yang [49] and Hirschfeld [24]. The correspondence
between Lie groups and Lie algebras was set up by Sophus Lie (1873–1893) in a purely
local way, i.e. between neighbourhoods of the identity in the Lie group and of zero
in the Lie algebra. Over half a century elapsed before the correspondence was made
global by Cartan, Pontryagin and Chevalley. A basic property of solvable Lie algebras