Number Theory: An Introduction to Mathematics

426 X A Character Study Lemma 16Ifρ:s→A(s)is a representation of degree n of a finite group G, then the characterχofρsatisfies | ...

7 Applications 427 Φn(x)= ∏ k (x−ωk) has integer coefficients, it follows thatα∈Q. Consequently, by Lemma 12,α∈Z. But|α|<1, s ...

428 X A Character Study It has been shown by Kazarin (1990) that the normal subgroup generated by the elements of the conjugacy ...

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 th ...

430 X A Character Study whereDis a diagonal matrix with positive diagonal elements. By the linear transfor- mationx=Tythe equati ...

7 Applications 431 thenVkis anmk-dimensional subspace ofRn(k= 1 ,...,p)andRnis the direct sum ofV 1 ,...,Vp. Moreover each eigen ...

432 X A Character Study Decomposing the characterχinto its irreducible components by means of (7), we obtainχ= 2 χ 1 + 2 χ 3. Si ...

8 Generalizations 433 The notions of left and right Haar integral obviously coincide if the groupGis abelian, and it may be show ...

434 X A Character Study thenL^1 (G)is aBanach algebraand M(f∗g)=M(f)M(g) for allf,g∈L^1 (G). Aunitary representationofGin a Hilb ...

8 Generalizations 435 For each fixeds∈G,themapˆs:χ→χ(s)is a character ofGˆ. Moreover the map s→ˆsis one-to-one, by the theorem o ...

436 X A Character Study the Fourier transform offis fˆ(t)= ∫∞ −∞ f(s)e−itsds, and the Fourier inversion formula has the form f(s ...

8 Generalizations 437 It may be shown that ifρis a unitary representation of a compact groupGin a Hilbert spaceH,thenH may be re ...

438 X A Character Study Moreover, every irreducible representation ofSU( 2 )is equivalent toρnfor some n≥0. To determine the cha ...

8 Generalizations 439 intimately related. This was already shown in§6 of Chapter I, but another version of the proof will now be ...

440 X A Character Study (iii) thegeneral linear group G L(n)of all real nonsingularn×nmatrices under matrix multiplication; (iv) ...

8 Generalizations 441 For every real Lie algebraL, there is a simply-connected Lie groupG ̃such that L(G ̃)=L. Moreover,G ̃is un ...

442 X A Character Study (i) a compact Lie group, and even any compact topological group, has only finitely many connected compon ...

9 Further Remarks 443 9 FurtherRemarks The history of Legendre’s conjectures on primes in arithmetic progressions is described i ...

444 X A Character Study was established by Lie, but we owe to Killing (1888–1890) the remarkable classifica- tion of simple comp ...

10 Selected References 445 [15] W. Ellison and F. Ellison,Prime numbers, Wiley, New York, 1985. [16] W. Feit,Characters of finit ...