# 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

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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).