Read e-book online Characters of Finite Groups. Part 2 PDF

By Ya. G. Berkovich and E. M. Zhmud

ISBN-10: 082184606X

ISBN-13: 9780821846063

This ebook discusses personality conception and its purposes to finite teams. The paintings areas the topic in the achieve of individuals with a comparatively modest mathematical history. the required history exceeds the normal algebra path with admire in simple terms to finite teams. beginning with uncomplicated notions and theorems in personality concept, the authors current quite a few effects at the houses of complex-valued characters and functions to finite teams. the most subject matters are levels and kernels of irreducible characters, the category quantity and the variety of nonlinear irreducible characters, values of irreducible characters, characterizations and generalizations of Frobenius teams, and generalizations and purposes of monomial teams. The presentation is special, and lots of proofs of identified effects are new. lots of the leads to the publication are offered in monograph shape for the 1st time. quite a few workouts provide additional info at the subject matters and aid readers to appreciate the most options and effects.

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Extra resources for Characters of Finite Groups. Part 2

Example text

27 IVµ, and D The estimate in Theorem 35 is probably the best possible. From now on, let us assume that n(~(G)) > 1. Then G/H is not a p-group (Lemma 8). As in Theorem 35, we will assume that all proper epimorphic images of G/H are abelian. Then (Exercise 13) G/H = (C(m),E(pn)). Fix 6 E lrr1(G/H) (by Exercise 13, 6(1) = m). Let X(m) be the component of ~that contains 6. Let X(p) = {x E Irr1(G) IP I x(l)}. 1 LEMMA 36. (a) X(m) (b) d(X(m)) $ 3. n X(p) = 0, X(m) U X(p) = lrr1(G). PROOF. Let X E lrr1(G) - {X(m) U X(p)}.

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

By assumption, IG: Hl¢(h) = ¢ 0 (h) = IHl-l L xEG ~(hx) = IHl-l L

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Characters of Finite Groups. Part 2 by Ya. G. Berkovich and E. M. Zhmud

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