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Representations and Characters of Groups, Second Edition by Martin Liebeck, Gordon James

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Applications to group theory

There are several ways of using the character theory of a group to determine information about the structure of the group. The examples which we have come across so far – finding the centre of the group, seeing whether or not the group is simple, and so on – require little calculation. In this chapter we present some rather deeper applications. The first involves doing arithmetic with character values to determine certain numbers, known as the class algebra constants. These constants carry information about the multiplication in G, and they can be used to investigate the subgroup structure of G, as we shall demonstrate. The second application takes this much further: the Brauer–Fowler Theorem 23.19 motivates the ...

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