Finite Permutation Groups by Helmut Wielandt, Henry Booker, D. Allan Bromley, Nicholas

By Helmut Wielandt, Henry Booker, D. Allan Bromley, Nicholas DeClaris

Finite Permutation teams by means of Helmut Wielandt Translated from the German by way of R. Bercov

Show description

Read or Download Finite Permutation Groups PDF

Similar mathematics_1 books

Arithmétique et travaux pratiques cycle d'observation classe de sixième

Manuel de mathématiques, niveau sixième. Cet ouvrage fait partie de los angeles assortment Lebossé-Hémery dont les manuels furent à l’enseignement des mathématiques ce que le Bled et le Bescherelle furent à celui du français.

Extra resources for Finite Permutation Groups

Example text

13. Primitive Groups with Transitive Subgroups 33 We prove this theorem by induction on | Δ \. For | Δ \ = 1, the assertion is trivial. Let | Δ | > 1. (a) First, we assume that 2| Δ \ < \ Ω |. This implies 2\Γ\ > | Ω |, and therefore for every geG we have ΓηΓ9 Φ 0 . Let α, βΕΔ. 1. Because Γ r\ Γ9 Φ 0 , H = (GA >g~lGAg) is transitive on Γν J \ since GA is transitive on Γ and g~*GAg is transitive on Γ9. Further, Δ = Δ η Δ9 is the set of all points which are left fixed by every element of H. Because a e i we have 1 < | Δ |, and because β $ Δ we have | Δ \ < \ Δ |.

Every \-fold transitive group is primitive br a Frobenius group. Proof. , let Gx = H have only orbits of length m > 1. Let us assume that G has a block of length k with 1 < k < n = \ Ω |. , n may be arranged in a matrix ai3 in such a way that every row φί is a block of G with length k and, say, « u = 1. φχ is fixed by Hf therefore k = l(w), hence (ky m) = 1. For i > 1, φ? is fixed by / / and does not contain 1. Therefore | φ? | is divisible by my but also by ky therefore by km. On the other hand, | α^ | = m, and therefore | φ?

1. In particular (n — l)-fold transitivity of G follows for m = 2 and (n — 2)-fold transitivity for m = 3 . Hence we have the following theorem. 3. A primitive group which contains a transposition is a symmetric group. A primitive group which contains a 3-cycle is either alternating or symmetric. For certain values of w, there exist groups G of degree n which contain transitive subgroups of degrees < n — 2 and are doubly but not triply transitive. These Jordan groups can be interpreted as automorphism groups of block designs (Hall, 1962).

Download PDF sample

Rated 4.89 of 5 – based on 31 votes