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

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22

Algebraic integers

Among the properties of characters which may be regarded as fundamental, perhaps the most opaque is that which states that the degree of an irreducible character of a finite group G must divide the order of G. This is one of several results which we shall prove in this chapter, using algebraic integers. Most of the results concern arithmetic properties of character values. We discuss properties of a group element gG which ensure that χ(g) is an integer for all characters χ of G. And we prove some useful congruence properties; for example, if p is a prime number and gG is an element of order pr for some r, then χ(g) χ(1) mod p for any character χ of G for which χ(g) is an integer.

Algebraic integers

22.1 Definition ...

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