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

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29

Permutations and characters

We have already seen in Chapter 13 that if G is a permutation group, i.e. a subgroup of Sn for some n, then G has a permutation character π defined by π(g) = |fix(g)| for gG, a fact which proved useful in many of our subsequent character table calculations. In this chapter we take the theory of permutation groups and characters somewhat further, and develop some useful results, particularly about irreducible characters of symmetric groups (see Theorem 29.12 below).

Group actions

We begin with a more general notion than that of a permutation group. If Ω is a set, denote by Sym(Ω) the group of all permutations of Ω. In particular, if Ω = {1, 2, . . . , n} then Sym(Ω) = Sn.

Definition

Let G be a group and Ω a set. ...

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