By Helmut Wielandt, Henry Booker, D. Allan Bromley, Nicholas DeClaris
Finite Permutation teams by way of Helmut Wielandt Translated from the German through R. Bercov
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If, in addition, K is doubly primitive and Γ = Γ', then either K is faithful or §18. Structure of the Constituents of Ga 49 m = t(t — 1). In the latter case L is imprimitive and has K as a first primitive component (W. A. Manning, 1927, 1929). 8. 5). §18. The Structure of the Transitive Constituents of G a in Primitive Groups G Throughout this section let G be primitive, and let Γ be an orbit of G a different from ex. 1. Let the subgroup V Φ 1 of G leave a point fixed. Then there is a g e G with g~xVg = U < G a and Ur φ 1.
A submodule 6 of W(H) is the transitivity module of a group G on H to be chosen appropriately, if and only if β has a basis OLX , ···, ocm , in which each at is a simple quantity of sJl(H) and m £<*,· = #. i=l Definition. We will call such a submodule of %H) S-module over H. an §22. Computation in S-Modules From the defining property of S-modules, namely, the existence of a basis of simple quantities, four remarks follow immediately, of which the first three are due to Schur (expressed differently; Schur avoided the use of the group ring).
Its fundamental idea consists in taking as * 'points" not, as we have done up to now, arbitrary objects or the numbers from 1 to 72, but rather group elements. §20. Introduction of Group Elements as Points Differing from Schur, who started with concepts from representation theory which will be discussed in the next chapter, we proceed in the following manner. If a permutation group G on Ω is to be investigated, we choose a regular permutation group H on Ω. It would not have to be contained in G, but this is the most important case, and we will assume H < G throughout.