Download e-book for kindle: Abstract harmonic analysis, v.2. Structure and analysis for by Edwin Hewitt, Kenneth A. Ross

By Edwin Hewitt, Kenneth A. Ross

ISBN-10: 0387048324

ISBN-13: 9780387048321

ISBN-10: 0387583181

ISBN-13: 9780387583181

ISBN-10: 3540048324

ISBN-13: 9783540048329

ISBN-10: 3540583181

ISBN-13: 9783540583189

This e-book is a continuation of vol. I (Grundlehren vol. one hundred fifteen, additionally to be had in softcover), and includes a precise remedy of a few very important elements of harmonic research on compact and in the neighborhood compact abelian teams. From the reports: ''This paintings goals at giving a monographic presentation of summary harmonic research, way more entire and accomplished than any publication already current at the reference to each challenge taken care of the ebook bargains a many-sided outlook and leads as much as most up-to-date advancements. Carefull consciousness is additionally given to the heritage of the topic, and there's an in depth bibliography...the reviewer believes that for a few years to come back this may stay the classical presentation of summary harmonic analysis.'' Publicationes Mathematicae

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Additional info for Abstract harmonic analysis, v.2. Structure and analysis for compact groups

Example text

Recall that Irr(t)(G) = {x E Irr(G) I x(l) = t}. 20. The following assertions hold for a (*)-group G with n > 0. 91, kerx1 for any X1 E lrru1 >(G). (b) ker '19 is solvable. (c) G' < G. (d) '19 vanishes on G - G', all prime divisors of IG: G'I divided, and (Xi)G' E Irr(G'), where Xi E Irrch>(G). (e) The group Lin(G) acts by multiplication on the set Irrui)(G), i E {1, ... , n}, semiregularly. LEMMA PROOF. Throughout the proof let Xi denote an irreducible character of G of degree fi, i E {1, ... , Xi E IrrcM(G)).

PROOF. Indeed, let K be the intersection of the kernels of the extensions to G of all¢ E Irr(H/~(H)), and let M/K be a 11'(H)'-Hall subgroup of the abelian group G/K. Then Mn H = ~(H), so that Mn P = ~(P) for all PE Syl(H). 52, M contains a normal p-complement for any p E 11'(H), and the intersection of all these complements is the normal complement of H in G. D THEOREM 10. Let H be a nilpotent Hall subgroup of G. Assume that for any ¢ E Irr#(H/~(H)), the degrees of all the nonlinear irreducible components of

Then ('¢x)(x) = '¢(x)[A1(x) + · · · + Am(x)] = m'¢(x) = '¢(x)x(l); hence '¢(x)(x(x) - x(l)) = O. Since xis faithful and x =/:- 1, it follows that x(x) =Im = x(l), and hence '¢(x) = 0. Therefore,"¢ vanishes on (G')#. This means that ¢G' is a multiple of PG', the regular character of G'. In particular, lG' E Irr('¢G'). So by Clifford's Theorem, ¢G' = ¢(1)1G'· Thus G' ::; ker'¢ = {1}. Then G is abelian, a contradiction. Thus we have proved the following result. (ii) No character in Irr(n) (G) is faithful.

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Abstract harmonic analysis, v.2. Structure and analysis for compact groups by Edwin Hewitt, Kenneth A. Ross

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