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

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

This ebook discusses personality concept and its purposes to finite teams. The paintings locations the topic in the succeed in of individuals with a comparatively modest mathematical historical past. the required historical past exceeds the traditional algebra path with recognize purely to finite teams. beginning with uncomplicated notions and theorems in personality thought, the authors current numerous effects at the houses of complex-valued characters and functions to finite teams. the most topics 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 plenty of proofs of recognized effects are new. many of the ends up in the publication are offered in monograph shape for the 1st time. a variety of workouts supply more information at the issues and aid readers to appreciate the most thoughts and effects.

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Extra resources for Characters of Finite Groups. Part 1

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If XT = Xu, then Trv U. Let mi= (xr,Xi) = (xu,x), i E {1, ... ,r}. By Corollary 10, T rv m1T1 + rv U. Hence T rv U. D Theorems 3 and 11 show that characters separate representation classes. Let T be a representation of a group G with matrix elements {aij}. Then the representation T of G with matrix elements {aij} is called the complex conjugate of T. Since xT(g) = xT(g) for all g E G, the character x = xT is completely determined by x = xr. It is called the complex conjugate of X· Using Lemma 7, we get x(g) = x(g) = x(g- 1).

It follows from the First Orthogonality Relation that X(G) E GL(k(G),q. We shall always assume that K 1 = {1}. Then the entries of the first column X1(G) of X(G) are the degrees of the irreducible characters of G (counting multiplicities). What kind of information about G can be derived from a knowledge of X(G)? Since IGI = ExEirr(G) x(1) 2 , it is evident that X(G) (even X1(G)) determines IGI, the order of G. Henceforth, we shall sometimes write Ex instead of ExEirr(G)' However, we shall not use this abbreviation when dealing with groups other than G.

If X1, X2 E Lin( G), then L:; 9 X1 (g h2 (9) = IGl8x 1,x2 • 2. If 91,92 E G, then 'L x(91)x(92) = IGl891,92· xELin(G) EXERCISE 25. Any linear character of a subgroup H of an abelian group G can be extended in IG: HI ways to a linear character of G. For a group G (not necessarily abelian) we define Lin(G) as the set of all homomorphisms of G into C*, and define multiplication in Lin( G) as in the abelian case. This definition makes Lin( G) a group. BASIC CONCEPTS 1. 16 THEOREM 14. Let G be a group (not necessarily abelian}, G' the derived group of G.

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