Get Character Theory of Finite Groups PDF

By I. Martin Isaacs

ISBN-10: 0123745500

ISBN-13: 9780123745507

First-class textual content ways characters through earrings (or algebras). as well as recommendations for employing characters to "pure" staff thought, a lot of the e-book makes a speciality of houses of the characters themselves and the way those houses replicate and are mirrored within the constitution of the crowd. difficulties stick with each one bankruptcy. Prerequisite a first-year graduate algebra direction. "A excitement to read."—American Mathematical Society. 1976 variation.

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This is, in fact, not hard to prove for group algebras over any field. We shall leave the general situation to the problems. 1) THEOREM Let C[G]-modules V and W afford characters x and $, respectively. Choose bases in V and W and construct V 0 W . Then V 0 W affords the character p j and is independent of the choice of bases. Proof Let {ui I 1 Ii In} and { w, I 1 Ir Irn} be bases for V and W , respectively, and let g E G . Write m ... n uig = x a i j u j w,g and = j= 1 with a i j ,brsE C. Then x(g) = by V 0 W .

Therefore IrrdG) = Irr(G). The point of this digression is to suggest that there is something “absolute” about a character table. It is not entirely an artifact of our choice of the particular field C. Another consequence of this argument which is sometimes useful is that if x E Irr(G),then x is afforded by an E-representation of G . This type of consideration will be discussed much more fully in Chapter 9. ” Group representations and characters 23 A great deal of information about a group can be recovered from its character table.

28 that Z(G) can be located from the character table of G. It follows that it can be determined from the table whether or not G is nilpotent. This is done by finding Z(G), then finding the character table of G/Z(G), and iterating this process. The sequence of subgroups of G which results is the upper central series and G is nilpotent iff this sequence reaches G. 27(c). We need a lemma first. 29) LEMMA Let H E G and let x be a character of G. Then I IG:HICx,xI with equality iff x(g) = 0 for all g E G - H.

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Character Theory of Finite Groups by I. Martin Isaacs


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