Download Character Theory of Finite Groups by I. Martin Isaacs PDF

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By I. Martin Isaacs

Very good textual content ways characters through jewelry (or algebras). as well as concepts for using characters to "pure" team conception, a lot of the ebook makes a speciality of homes of the characters themselves and the way those homes replicate and are mirrored within the constitution of the crowd. difficulties stick with each one bankruptcy. Prerequisite a first-year graduate algebra path. "A excitement to read."—American Mathematical Society. 1976 version.

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Since H E Z[G], we have Z[H]c Z[G]. We show that G E Z[XJ. WriteG = { g i l l 5 i In}andH = { h i l l Ii In},wheren = IGI = IHI. Chapter 3 44 xi We have hi = aijgj for aijE Z. We shall show that the matrix A = (aij) has an inverse with entries in Zand this will complete the proof. Write h j - = bijgi-l and B = (bij), an integer matrix. Now express h i h j - ' as a linear combination ofelements of G and observe that the coefficient of 1 is a i , b V j It . follows that xi cy nsij = p(hihj-') = n c V and thus A B = I, the identity matrix.

The reader should check that if u E V ,w E W , and g E G, then ( u 0 w)g = ug @I wg. It follows that (xgl)gz = x(g,g,) for x E V 0 W and g i E G. Next we give V 0 W the structure of a C[G]-module by extending the action of G by linearity in @[GI. In other words, for x E V 0 W we define c a&) = a , ( w ) . It is routine to check that this really makes V 0 W into a C[G]-module. 41 Chapter 4 48 A few words of caution are in order here. If a E C[G], it is not necessarily true that (ui 0 wj)a = via 0 wja.

Furthermore, under this correspondence, irreducible representations correspond to irreducible representations. This situation may be interpreted in terms of characters as follows. 22) LEMMA Let N 4 G. (a) If x is a character of G and N E ker x, then x is constant on cosets of N in G and the function 2 on GIN defined by f ( N g ) = x(g) is a character of GIN. (b) If 2 is a character of G I N , then the function x defined by x(g) = f(Ng) is a character of G. (c) In both (a) and (b), x E Irr(G) iff 2 E Irr(G/N).

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