Characters of Finite Groups. Part 1 by Ya. G. Berkovich and E. M. Zhmud

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

This e-book discusses personality idea and its purposes to finite teams. The paintings areas the topic in the succeed in of individuals with a comparatively modest mathematical historical past. the mandatory heritage exceeds the normal algebra path with admire merely to finite teams. beginning with simple notions and theorems in personality thought, the authors current a number of effects at the houses of complex-valued characters and purposes to finite teams. the most issues 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 functions of monomial teams. The presentation is designated, and lots of proofs of identified effects are new. lots of the ends up in the publication are awarded in monograph shape for the 1st time. quite a few workouts supply additional info at the issues and support readers to appreciate the most options and effects.

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Let i : H :::; G be an embedding, T : G---+ GL(n, F) a representation. Then TH= Toi is the restriction of T to H, which maps any x EH into T(x). THEOREM 8. Let T be a faithful irreducible matrix representation of a group G over an algebraically closed field F, and let H :::; G. If TH is irreducible then Ca(H) = Z(G). Proof. Let n =deg T. Then, by Exercise 16, CcL(n,F)(T(H)) consists of scalar matrices. Let g E Ca(H). Then T(g) is scalar. , Ca(H) :::; Z(G). The opposite inclusion is obvious. D §8.

LEMMA 4. Let A be an algebra over a field F, and V, V' irreducible A-modules. Then if V ~ V', Hom A (V, V') = { skew field if V = V'. ~ Moreover, if Fis algebraically closed, then EndA(V) = F · idv. Proof. Let 0 =f. ¢ E HomA(V, V'). Since ker¢ and im ¢ are submodules of V and V', respectively, it follows that ker ¢ = 0 and im ¢ = V'. Consequently, ¢ is an isomorphism of V onto V'. Let V = V'. Then, as we now know, HomA (V, V) = EndA (V) is a skew field. Suppose that F is algebraically closed. Then the characteristic polynomial

X This enables us to view V as an FG-module. Identifying an element g of G with the element Lx Dx,gX of FG, we may consider Gas a subset of FG. It is clear that §5. MATRIX REPRESENTATIONS 7 any FG-module can be considered as a G-module and, conversely, any G-module can be considered as an FG-module. In what follows, no distinction will be made between these two types of modules. If T : G -+ GL(V) is a representation, we get a representation of ·the algebra FG by setting r(a) = L:a(x)r(x) (a EFG), x which will be denoted by the same letter.

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