Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Chap. 0 Preliminaries
- Chap. I Solvable subgroups of linear groups
- Chap. II Solvable permutation groups
- Chap. III Module actions with large centralizers
- Chap. IV Prime power divisors of character degrees
- Chap. V Complexity of character degrees
- Chap. VI π-special characters
- References
- Index
Chap. II - Solvable permutation groups
Published online by Cambridge University Press: 23 September 2009
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Chap. 0 Preliminaries
- Chap. I Solvable subgroups of linear groups
- Chap. II Solvable permutation groups
- Chap. III Module actions with large centralizers
- Chap. IV Prime power divisors of character degrees
- Chap. V Complexity of character degrees
- Chap. VI π-special characters
- References
- Index
Summary
Orbit Sizes of p-Groups and the Existence of Regular Orbits
Let G be a permutation group on a finite set Ω. The orbit {ωg | g ∈ G} is called regular, if CG(ω) = 1 holds.
In this section we consider a finite p-group P which acts faithfully and irreducibly on a finite vector space V of characteristic q ≠ p. For several questions in representation theory, it turns out to be helpful if one knows that P has a long orbit (preferably a regular orbit) in its permutation action on V. For applications, see §14.
We start with an easy, but useful, lemma.
Lemma. Let G act on a vector space V over GF(q), and let Δ ≠ 0 be an orbit of G on V. If λ ∈ GF(q)#, then λΔ ≠ Δ or o(λ) | exp(G).
Proof. Let λΔ = Δ. Then, for ν ∈ Δ, there exists g ∈ G such that λν = νg. If n = o(g), then ν = νgn = νλn and o(λ) | n.
We recall that and denote the set of Fermat and Mersenne primes, respectively.
Lemma. Let P be a non-trivial p-group and V a faithful, irreducible and primitive P-module over GF(q) for a prime q ≠ p. Set |P| = pnand |V|= qm.
(a) There always is a regular orbit of P on V, except the case where and P is dihedral or semi-dihedral. In this exceptional case, clearly P has D8 ≅ Z2wrZ2 as a subgroup.
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- Representations of Solvable Groups , pp. 73 - 116Publisher: Cambridge University PressPrint publication year: 1993